GENERATION OF FUSION SYSTEMS OF CHARACTERISTIC 2-TYPE

Michael Aschbacher California Institute of Technology

Let p be a prime, G a finite group, and S a Sylow p-subgroup of G. Subsets of S are said to be fused in G if they are conjugate in G.

The study of fusion in Sylow subgroups of finite groups is an old and important topic in finite group theory. Perhaps the most basic theorem in the subject is Alperin’s Fusion Theorem [Al], which says fusion in S is controlled in p-local subgroups H of G such that S ∩ H ∈ Syl_{p}(H) and C_{H}(O_{p}(H)) ≤ O_{p}(H). See Glauberman’s lectures in [Gl2] for other results on fusion and more references on the subject. As is apparent from [Gl2], a reoccurring theme in the study of fusion is the search for some small set of characteristic subgroups of S whose normalizers control fusion in S. Our strongest theorem is of this flavor.

Define G to be of characteristic p-type if for each p-local subgroup H of G, C_{H}(O_{p}(H)) ≤ O_{p}(H). In this paper we investigate 2-fusion in groups of characteristic 2-type. Recall that the proof of the theorem classifying the finite simple groups proceeds by partitioning the simple groups into two classes: those of characteristic 2-type, and those not of characteristic 2-type. Thus the groups of characteristic 2-type are of great interest. See the expository article [A1] for a discussion of the role of groups of characteristic 2-type in the classification theorem.

Actually we consider fusion in the larger context of saturated fusion systems, and the strongest versions of our theorems are results about fusion systems rather than groups. These objects were defined and first studied by Puig, although Puig calls them Frobenius categories rather than fusion systems; see in particular [P1] and [P2]. Our introduction to the subject was from [BLO], and we adopt the notation and terminology found there.

We now introduce some definitions necessary to state our main theorems. The reader is directed to [FGT] for notation and terminology involving finite groups, and to Section 1 and Appendix A of [BLO] for notation and terminology involving fusion systems. Section 2 of this paper also contains some background on fusion systems.

This work was partially supported by NSF-0504852

Let p be a prime and F a saturated fusion system on a finite p-group S. We say that F is of characteristic p-type if for each nontrivial fully normalized subgroup U of S, N_{F}(U) is constrained. For example if G is a finite group of characteristic p-type and S ∈ Syl_{p}(G), then the standard fusion system F_{S}(G) (encoding fusion in S induced in G) is of characteristic p-type. The converse is not true however; as we will see, there are groups G which are not of characteristic p-type, but have F_{S}(G) of characteristic p-type. This is just one example showing that some theorems about finite groups have more attractive statements and proofs in the language of fusion systems.

A finite group H is a K-group if each simple section of H is on the list K of “known” simple groups appearing in the statement of the theorem classifying the finite simple groups. Define F to be a local CK-system if for each nontrivial subgroup P of S, Aut_{F}(P) is a K-group. For example if G is a minimal counter example to the classification and S ∈ Syl_{p}(G), then F_{S}(G) is a local CK-system. Since our results can be used to simplify portions of the proof of the classification theorem, it is important not to use that theorem in their proofs, but instead to prove theorems about local CK-systems.

Assume p = 2 and define F to be an obstruction to pushing up at the prime 2 if F = F_{S}(G) for some finite group G with S ∈ Syl_{2}(G) such that one of the following holds:

(1) S is dihedral of order at least 16 and G ≅ L_{2}(q) or P GL_{2}(q) for some odd prime power q.

(2) S is semidihedral and G ≅ L_{2}(q)^{(1)} for some odd prime power q which is a square.

(3) S is semidihedral of order 16 and G ≅ L_{3}(3).

(4) |S| = 32 and G ≅ Aut(A_{6}) or Aut(L_{3}(3)).

(5) |S| = 2^{7} and G ≅ J_{3}.

(6) F^{∗}(G) ≅ Sp_{4}(q), where q = 2^{e} with e > 1 odd, |G : O^{2}(G)| = 2, and O^{2}(G) is an extension of F^{∗}(G) by a group of field automorphisms of odd order.

(7) F^{∗}(G) ≅ L_{3}(q), q = 2^{e}, e > 1, |O^{2}(G) : F^{∗}(G)| is odd, and G is the extension of O^{2}(G) by a graph or graph-field automorphism.

Here L_{2}(q)^{(1)} is the extension of L_{2}(q) of degree 2 with semidihedral Sylow 2-subgroups. We can now state our first theorem:

Theorem 1. Assume F is a saturated fusion system on a finite 2-group S, such that F is a local CK-system of characteristic 2-type. Let U be the set of nontrivial normal

GENERATION OF FUSION SYSTEMS OF CHARACTERISTIC 2-TYPE 3

subgroups U of S such that C_{S}(U) ≤ U and O_{2}(Aut_{F}(U)) = Inn(U). Then either (1) F = ⟨N_{F}(U) : U ∈ U⟩ is generated by the normalizers of members of U, or

(2) F is an obstruction to pushing up at the prime 2.

As a corollary to Theorem 1, we obtain the corresponding theorem for groups:

Theorem 2. Assume G is a finite group of characteristic 2-type in which all 2-local subgroups are K-groups, and let S ∈ Syl_{2}(G). Then one of the following holds:

(1) The subgroups N_{G}(U), such that C_{S}(U) ≤ U ⊴ S and U = O_{2}(N_{G}(U)), control fusion in S.

(2) S is dihedral and G ≅ L_{2}(p), where p is a Fermat or Mersenne prime.

(3) S is semidihedral of order 16 and G ≅ M_{10}, M_{11}, or L_{3}(3).

(4) |S| = 32 and G ≅ Aut(L_{3}(3)).

(5) |S| = 2^{7} and G ≅ J_{3}.

Observe the list of obstructions in Theorem 2 is smaller than the corresponding list in Theorem 1. This is because most of the groups G, appearing in the list of obstructions to pushing up at the prime 2, are not of characteristic 2-type, even though their fusion systems F_{S}(G) are of characteristic 2-type.

Here is another corollary to Theorem 1:

Corollary 3. Assume F is a saturated fusion system on a finite 2-group S, such that F is a local CK-system of characteristic 2-type. Partially order the set U of Theorem 1 by U ≲ V if V ≤ U and N_{F}(U) ≤ N_{F}(V), and assume U has a greatest member U. Then there exists a finite group G with S ∈ Syl_{2}(G) such that F = F_{S}(G), and one of the following holds:

(1) F is constrained and U = O_{2}(F) = O_{2}(G).

(2) S is dihedral of order at least 16, U = S, and G ≅ L_{2}(q) or P GL_{2}(q) for some odd prime power q.

(3) S is semidihedral, U = S, and G ≅ L_{2}(q)^{(1)} for some odd prime power q which is a square.

(4) S is semidihedral of order 16, U = O_{2}(C_{G}(Z(S)) ≅ Q_{8}, and G ≅ L_{3}(3).

(5) |S| = 32, G ≅ Aut(A_{6}) or Aut(L_{3}(3)), and U = S or O_{2}(C_{G}(Z(S)), respectively.

(6) F^{∗}(G) ≅ Sp_{4}(q), where q = 2^{e} with e > 1 odd, U = Baum(S), |G : O^{2}(G)| = 2, and O^{2}(G) is an extension of F^{∗}(G) by a group of field automorphisms of odd order.

(7) F^{∗}(G) ≅ L_{3}(q), q = 2^{e}, e > 1, |O^{2}(G) : F^{∗}(G)| is odd, and G is the extension of O^{2}(G) by a graph or graph-field automorphism. Either U = Baum(S) or q = 4, G is the extension of L_{3}(4) by a graph automorphism, and U = S.

Recall the Thompson subgroup J(S) is the subgroup of S generated by the elementary abelian subgroups of S of maximal order, while the Baumann subgroup Baum(S) of S is defined by Baum(S) = Ω_{1}(Z(C_{S}(J(S)))).

Here is a version of Corollary 3 for groups:

Corollary 4. Assume G is a group of characteristic 2-type in which all 2-locals are K-groups, and S ∈ Syl_{2}(G) is contained in a unique maximal 2-local subgroup M of G. Then one of the following holds:

(1) G is of 2-rank 1, M = S, and G = O(G)S.

(2) G/O(G) ≅ SL_{2}(3) or SL_{2}(3)^{(2)}, and M = C_{G}(Z(S)).

(3) O_{2}(G) ≠ 1 and M = G.

(4) S is dihedral of order at least 16, M = S, and G ≅ L_{2}(p), where p is a Fermat or Mersenne prime.

(5) S is semidihedral of order 16, M = C_{G}(Z(S)), and G ≅ M_{10}, M_{11} or L_{3}(3).

(6) |S| = 32, G ≅ Aut(L_{3}(3)), and M = C_{G}(Z(S)).

(7) G is a simple group of Lie type of even characteristic and Lie rank 1, extended by a group of field automorphisms of odd order, and M = N_{G}(S).

Here SL_{2}(3)^{(2)} is the extension of SL_{2}(3) of degree 2 with quaternion Sylow 2subgroups.

The situation where S is in a unique maximal 2-local is an interesting and difficult case in the analysis of groups of characteristic 2-type; see for example Chapter 2 in [ASm], which handles this case when G is quasithin. Unfortunately Corollary 4 is not strong enough for the purposes of [ASm], as the groups there are of even characteristic, rather than of characteristic 2-type. However the result does suffice for purposes of the original approach to the classification of the finite simple groups.

Theorem 1 has many other consequences. For example, in a later paper, we use Theorem 1 to prove all S_{3}-free saturated fusion systems are constrained. Then we use that result in turn to give a new and very simple treatment of S_{4}-free finite groups, which were first classified by Glauberman in [Gl2].

Our last and strongest theorem shows that, with known exceptions, if F is a 2fusion system of characteristic 2-type, then the normalizers of two naturally defined F-characteristic subgroups of S control fusion in F.

Theorem 5. Assume F is a saturated fusion system on a finite 2-group S, such that F is a local CK-system of characteristic 2-type. Then one of the following holds:

(1) F = ⟨C_{F}(Ω_{1}(Z(S))), N_{F}(Baum(S))⟩.

(2) G(F) ≠ ∅ and for each Campbell pair (C_{1}, C_{2}) for F, F = ⟨C_{F}(C_{1}), N_{F}(C_{2})⟩.

(3) There exists a finite group G with S ∈ Syl_{2}(G) such that F = F_{S}(G) and either G is M_{23} or G is an extension of L_{3}(2^{e}) or Sp_{4}(2^{e}) by a group of odd order for some positive integer e.

(4) F is an obstruction to pushing up at the prime 2.

The collection G(F) is defined in Definitions 8.8 and 1.3, and the notion of a Campbell pair is defined in Definition 8.8. A Campbell pair is a pair (C_{1}, C_{2}) of subgroups of S, defined fairly explicitly in terms of some group in G(F), such that C_{1} ≤ Ω_{1}(Z(S)) is Aut_{F}(S)-invariant, and C_{2} ≤ Baum(S) is Aut_{F}(Baum(S))-invariant.

We hope that Theorem 5 will serve as a first step in a program to classify the saturated 2-fusion systems of characteristic 2-type.

The proofs of our various theorems draw heavily on machinery developed or collected in [ASm]. Many of the lemmas in the proof are minor variations on lemmas about groups found in [ASm], slightly modified to apply to fusion systems. On the other hand, many of these arguments seem to be easier and more attractive when carried out in the category of fusion systems rather than the category of groups.

McBride’s notion of a “minimal parabolic” (cf. Definition 1.1) plays an important role in the proof, as does Campbell’s approach in [C] to the Glauberman-Niles Theorem [GN].

It would be nice to extend our theorems in at least two directions. First, and most obviously, one should presumably prove versions of the theorems for all primes, rather than just the prime 2. This would entail extending some of the results in [ASm] for the prime 2 to all primes. For the most part, such extensions should not be difficult, and in many cases may already exist. The prime 3 would present some special difficulties, as can be seen from [GN], [N], and the existence of G_{2}(3^{n}).

But more important, one would like to remove the hypothesis that the fusion system or group is of characteristic 2-type, presumably at the expense of adding more obstructions. Since Alperin’s Fusion Theorem tells us that fusion is controlled in constrained locals, it does not seem impossible that one could treat the case where F or G is not of characteristic 2-type. Section 1. GN-pairs

In this section S is a finite 2-group. Recall A(S) denotes the set of elementary abelian subgroups of S of maximal order.

We begin by recalling the definition of a minimal parabolic in a finite group from B.6.1 in [ASm]; the notion is due to P. McBride. Definition 1.1. Let G be a finite group, p a prime, and P ∈ Syl_{p}(G). A minimal parabolic of G over P is an overgroup H of P in G such that P is contained in a unique maximal subgroup of H, and P is not normal in H.

Theorem 1.2. (McBride) Let G be a finite group, p a prime, and P ∈ Syl_{p}(G).

(1) If G = O^{p′}(G) and P is not normal in G, then G is generated by the minimal parabolics over P.

(2) G is generated by N_{G}(P) and the minimal parabolics over P.

Proof. See B.6.5 in [ASm].

Next we recall Definition C.1.17 from [ASm]: Definition 1.3. Denote by G(S) the set of finite groups H such that S ∈ Syl_{2}(H), F^{∗}(H) = O_{2}(H), H is a minimal parabolic over S, and setting W = Ω_{1}(Z(O_{2}(H))) and H^{∗} = H/C_{H}(W), we have F^{∗}(H^{∗}) ≅ L_{2}(2^{n})′ with [W, F^{∗}(H^{∗})]/C_{[W,F^{∗}(H^{∗})]}(F^{∗}(H^{∗})) the natural module for F^{∗}(H^{∗}). Set τ(H) = (H, F, V) where F = Ω_{1}(Z(J(O_{2}(H)))) and V = [O^{2}(G), F]. Define T (S) = {τ(H) : H ∈ G(S)}.

(1.4) Assume S = Baum(S) and (H, F, V) ∈ T (S). Set Z = Ω_{1}(Z(S)), W = ⟨Z^{H}⟩, H^{∗} = H/C_{H}(W), Q = O_{2}(H), and K = O^{2}(H). Let M be the unique maximal overgroup of S in H and J = ker_{M}(H). Then

(1) Q = O_{2}(J) ∈ Syl_{2}(J) and J = C_{H}(W).

(2) W = V C_{Z}(H) so Q = C_{S}(V).

(3) V/C_{V} (H) is the natural module for H^{∗} ≅ L_{2}(2^{n}).

(4) Z = C_{V} (S)C_{Z}(H).

(5) For each A ∈ A(S), F = V C_{F} (A) = V (F ∩ A) and V (A ∩ Q) ∈ A(S), so A(Q) ⊆ A(S).

(6) J(S) ≰ Q and for each A ∈ A(S) − A(Q), S = AQ, m(A/A ∩ Q) = n = m(V/(A ∩ V), and C_{V} (S) = [V, A].

Proof. From 1.3, H is a minimal parabolic over S, F^{∗}(H^{∗}) ≅ L_{2}(2^{n})′, and

[W, F^{∗}(H^{∗})]/C_{[W,F^{∗}(H^{∗})]}(F^{∗}(H^{∗}))

is the natural module for F^{∗}(H^{∗}). Let L_{+} be the preimage in H of F^{∗}(H^{∗}), L = O^{2}(L_{+}), and set V_{W} = [W, F^{∗}(H^{∗})].

By parts (1) and (5) of B.6.8 in [ASm], Q = O_{2}(J) ∈ Syl_{2}(J), and by part (6) of that lemma, C_{H}(W) = J, so (1) is established.

If J(S) ≤ J then F = Ω_{1}(Z(J(S)), and then as S = Baum(S), we have Z = F. But then W = F = Z and S = C_{S}(Z) = C_{S}(W), contrary to the previous paragraph. Hence J(S) ≰ J. Then by B.6.9 in [ASm], K^{∗} = L^{∗}, so K = L. Moreover as J(S) ≰ J, the hypotheses of Baumann’s Lemma (cf. B.6.10 in [ASm]) are satisfied, and by that lemma, Z = C_{V_{W}} (J(S))C_{Z}(K).

Next as Z ≤ F, also W = ⟨Z^{H}⟩ ≤ F. Now W = [W, K]Z = V_{W} Z, so as Z = C_{V_{W}} (J(S))C_{Z}(K), W = V_{W} C_{Z}(H). As J(S) ≰ Q there exists A ∈ A(S) with A ≰ Q. By B.4.2.1 in [ASm], m(W/C_{W} (A)) = n and A^{∗} ∈ Syl_{2}(K^{∗}) if n = 1, while H^{∗} = A^{∗}K^{∗} ≅ L_{2}(2) if n = 1. Then by B.2.4.2 in [ASm], W (A ∩ Q) ∈ A(S). In particular A(Q) ⊆ A(S) and W (A ∩ Q) ∈ A(Q), so F ≤ W (A ∩ Q).

Recall from paragraph one that V_{W} /C_{V_{W}} (K) is the natural module for K^{∗} ≅ L_{2}(2^{n})′. In particular as either A^{∗} ∈ Syl_{2}(K^{∗}), or n = 1 and H^{∗} = A^{∗}K^{∗} = L_{2}(2), it follows that m(V_{W} /C_{V_{W}} (A)) = n. Then as m(W/C_{W} (A)) = n, W = V_{W} C_{W} (A) so W ≤ V_{W} (A ∩ Q) ∈ A(S) and C_{V_{W}} (A)/C_{V_{W}} (K) = C_{V_{W} /C_{V_{W}} (K)}(A^{∗}), so C_{V_{W}} (A) is the preimage V_{0} in V_{W} of C_{V_{W} /C_{V_{W}} (K^{∗})}(S_{0}^{∗}), where S_{0}^{∗} ∈ Syl_{2}(K^{∗}) when n = 1, and S_{0}^{∗} = S^{∗} if n = 1. Therefore V_{0} = C_{V_{W}} (J(S)), so V_{0} = Z ∩ V. Hence S = C_{S}(Z) centralizes V_{0}, so S^{∗} = S_{0}^{∗} and therefore H^{∗} = K^{∗}S_{0}^{∗} ≅ L_{2}(2^{n}). In particular A^{∗} ∈ Syl_{2}(H^{∗}), so A^{∗} = S^{∗} and hence S = AQ.

Next as W ≤ F ≤ W (A ∩ Q) and W = V_{W} (A ∩ W), F = V_{W} (A ∩ F), so A centralizes F/V_{W}. Therefore K = [K, A] centralizes F/V_{W}, so V_{W} = [K, F] = V. As V = V_{W}, various facts established above prove (2)-(5), and all parts of (6) except the statement that C_{V} (S) = [V, A], which is proved in Lemma 1.4 in [C].

Definition 1.5. A Baumann pair for S is a pair (C_{1}, C_{2}) of nontrivial characteristic subgroups of S such that C_{1} ≤ Ω_{1}(Z(S))) and C_{2} is characteristic in Baum(S). A Glauberman-Niles pair for S (GN-pair for S) is a Baumann pair (C_{1}, C_{2}) for S such that for each H ∈ G(Baum(S)), one of the follwing holds:

(1) C_{1} ≤ Z(H).

(2) C_{2} ⊴ H.

(3) O^{2}(H) is an L_{2}(2^{n})-block or A_{3}-block. See Definition C.1.7 in [ASm] for the definition of the notion of a block, and see Definition C.1.12 in [ASm] for the definition of an L_{2}(2^{n})-block and an A_{3}-block.

Theorem 1.6. (Glauberman-Niles) For each nontrivial finite 2-group S, there exists a Glauberman-Niles pair for S.

Proof. See [GN]. There is also a proof in [C] due to Campbell supplying a specific GNpair, which we sketch below.

Theorem 1.7. (Campbell) Let S = Baum(S) and assume (H_{i}, F_{i}, V_{i}) ∈ T (S) for i = 1, 2. If V_{1} ≤ O_{2}(H_{2}) then either F_{1} ≤ F_{2} or C_{V_{1}}(S) ≤ Z(H_{2}).

Proof. A somewhat stronger version of this result appears as Theorem B in [C].

Corollary 1.8. Assume S = Baum(S) and (H_{i}, F_{i}, V_{i}) ∈ T (S) for i = 1, 2. Then one of the following holds:

(1) F_{1} = F_{2}.

(2) C_{V_{i}}(S) ≤ Z(H_{3−i}) for i = 1 and 2.

(3) [V_{1}, V_{2}] ≠ 1.

Proof. Assume (3) fails and set Q_{i} = O_{2}(H_{i}). Then V_{3−i} ≤ C_{S}(V_{i}) = Q_{i} by 1.4.2. Therefore by 1.7:

(a) For i = 1 and 2, either F_{i} ≤ F_{3−i} or C_{V_{i}}(S) ≤ Z(H_{3−i}).

Suppose for i = 1 or 2 that F_{i} ≤ F_{3−i}. By 1.4.6, there is A ∈ A(S) − A(Q_{i}) and [V_{i}, A] = C_{V_{i}}(S). As V_{i} ≤ F_{i} ≤ F_{3−i} and [V_{i}, A] ≠ 1, also [F_{3−i}, A] ≠ 1. Indeed by 1.4.5, F_{3−i} = V_{3−i}C_{F_{3−i}}(A), so [F_{3−i}, A] = [V_{3−i}, A] = C_{V_{3−i}}(S). Therefore

C_{V_{i}}(S) = [V_{i}, A] ≤ [F_{i}, A] ≤ [F_{3−i}, A] = C_{V_{3−i}}(S),

so as C_{V_{i}}(S) ≰ Z(H_{i}), it follows that C_{V_{3−i}}(S) ≰ Z(H_{i}). Hence by (a), F_{3−i} ≤ F_{i}, so (1) holds.

We have shown that if neither (1) nor (3) holds, then for i = 1 and 2, F_{i} ≰ F_{3−i}. Hence (2) holds by (a), completing the proof of the lemma.

Theorem 1.9. (Campbell) Assume S = Baum(S) and (H, F, V) ∈ T (S). Let Inn(S) ≤ Λ ≤ Aut(S). Then either

(1) for each α ∈ Λ, [F^{α}, H] ≤ F, or

(2) for some α ∈ Λ, [V, V^{α}] ≠ 1, so that S = QV^{α}.

Proof. Again a stronger version of this result appears as Theorem A in [C].

Corollary 1.10. Assume S = Baum(S) and τ_{i} = (H_{i}, F_{i}, V_{i}) ∈ T (S) for i = 1, 2. Let Inn(S) ≤ Λ ≤ Aut(S). Define

C_{1} = C_{1}(τ_{1}, Λ) = ⟨C_{V_{1}}(S)^{Λ}⟩ and C_{2} = C_{2}(τ_{1}, Λ) = ⟨F_{1}^{Λ}⟩.

Then one of the following holds:

(1) C_{1} ≤ Z(H_{2}).

(2) C_{2} ⊴ H_{2}.

(3) For some α ∈ Λ, [V_{1}^{α}, V_{2}] ≠ 1.

Proof. Assume (3) fails and set Q_{i} = O_{2}(H_{i}). As in the proof of 1.8:

(a) For each α ∈ Λ, V_{1}^{α} ≤ Q_{2} and V_{2} ≤ Q_{1}^{α}. Next by 1.4.5, for A ∈ A(Q_{2}) and α ∈ Λ, F_{1}^{α} = V_{1}^{α}(A ∩ F_{1}^{α}), so by (a):

(b) For each α ∈ Λ, F_{1}^{α} ≤ Q_{2}. Further by (a) and 1.8:

(c) For each α ∈ Λ, either F_{1}^{α} = F_{2} or C_{V_{1}}(S)^{α} ≤ Z(H_{2}). Suppose for some β ∈ Λ that F_{1}^{β} = F_{2}. Then C_{2} = ⟨F_{1}^{Λ}⟩ = ⟨F_{2}^{Λ}⟩. Also by (b), for each

γ ∈ Λ, V_{2}^{γ} ≤ F_{2}^{γ} = F_{1}^{β^{−1}γ} ≤ Q_{2}, so by 1.9, F_{2}F_{2}^{γ} ⊴ H_{2}. Therefore C_{2} = ⟨F_{2}^{Λ}⟩ ⊴ G_{2}, so that (1) holds.

Therefore we may assume that for all β ∈ Λ, F_{1}^{β} ≠ F_{2}. Thus by (c), for all α ∈ Λ, C_{V_{1}}(S)^{α} ≤ Z(H_{2}). But then (1) holds, completing the proof of the corollary.

Compare the following lemma to C.5.6 in [ASm], which was in turn suggested in part by 3.4 in [St].

(1.11) Assume S = Baum(S) and τ_{i} = (H_{i}, F_{i}, V_{i}) ∈ T (S) for i = 1, 2, with [V_{1}, V_{2}] ≠ 1. Set Q_{i} = O_{2}(H_{i}), K_{i} = O^{2}(H_{i}), D_{i} = C_{S}(K_{i}), and E_{i} = [V_{i}, S]. Then there exists a positive integer n such that for i = 1 and 2:

(1) H_{i}/C_{H_{i}}(V_{i}) ≅ L_{2}(2^{n}) and K_{i} is an L_{2}(2^{n})-block or an A_{3}-block.

(2) Q_{i} = V_{i}D_{i} with D_{i} ≤ C_{Q_{i}}(V_{3−i}).

(3) S = Q_{i}V_{3−i} = Q_{i}Q_{3−i}, Q_{i} = V_{i}(Q_{i} ∩ Q_{3−i}), and E_{i} = C_{V_{i}}(S) = [V_{1}, V_{2}] = C_{V_{i}}(V_{3−i}) = V_{1} ∩ V_{2}.

(4) E_{1} = E_{2}.

(5) Φ(D_{1}) = Φ(D_{2}).

(6) Φ(D_{i}) ≤ D_{1} ∩ D_{2} and m(D_{i}/D_{1} ∩ D_{2}) ≤ n.

Proof. Set H_{i}^{∗} = H_{i}/C_{H_{i}}(V_{i}). First as H_{i} ∈ G(S), H_{i}^{∗} ≅ L_{2}(2^{n_{i}}) for some positive integer n_{i} by 1.4.3, and by 1.4.5, V_{i} ≤ A_{i} ∈ A(S). Then as [V_{1}, V_{2}] ≠ 1, [V_{i}, A_{3−i}] ≠ 1, so by 1.4.6, S = Q_{i}A_{3−i} , C_{V_{i}}(S) = V_{i} ∩ A_{3−i} = C_{V_{i}}(a) for a ∈ A_{3−i} − Q_{i}, and m(V_{i}/C_{V_{i}}(S)) = n_{i} = m(A_{3−i}/(A_{3−i} ∩ Q_{i})). In particular for v_{3−i} ∈ V_{3−i} − Q_{i},

n_{i} = m(V_{i}/C_{V_{i}}(v_{3−i})) = m(V_{i}/C_{V_{i}}(V_{3−i})) = m(V_{i}/(V_{i}∩Q_{3−i}) ≤ m(A_{i}/(A_{i}∩Q_{3−i})) = n_{3−i}.

It follows from symmetry between τ_{1} and τ_{2} that n_{1} = n_{2} = n and S = Q_{i}V_{3−i}. Indeed Q_{3−i} = C_{S}(V_{3−i}) is of index 2^{n} in S, and |V_{i} : C_{V_{i}}(V_{3−i})| = 2^{n}, so Q_{i} = V_{i}C_{Q_{i}}(V_{3−i}) = V_{i}(Q_{i}∩Q_{3−i}). Then [Q_{i}, V_{3−i}] ≤ V_{i}, so O^{2}(G_{i}) = [O^{2}(G_{i}), V_{3−i}] centralizes Q_{i}/V_{i}. Hence O^{2}(G_{i}) is a block, completing the proof of (1).

Next by 1.4.6, C_{V_{i}}(S) = [V_{i}, A_{3−i}], so as S = V_{3−i}Q_{i} = A_{3−i}Q_{i} and Q_{i} = C_{S}(V_{i}), it follows that C_{S}(V_{i}) = [V_{i}, S] = E_{i} = [V_{1}, V_{2}] = C_{V_{i}}(V_{3−i}). As V_{j} ⊴ S for j = 1, 2, [V_{1}, V_{2}] ≤ V_{1} ∩ V_{2}, and of course V_{1} ∩ V_{2} ≤ C_{V_{i}}(V_{3−i}), so also V_{1} ∩ V_{2} = [V_{1}, V_{2}]. This completes the proof of (3). Then (3) implies (4).

Let h_{i} ∈ H_{i} − N_{H_{i}}(S^{∗}) and set L_{i} = ⟨V_{3−i}, V_{3−i}^{h_{i}}⟩. Thus L_{i}^{∗} = H_{i}^{∗}. Next Q_{i} ≤ S ≤ N_{H_{i}}(V_{3−i}), so Q_{i} = Q_{i}^{h_{i}} also acts on V_{3−i}^{h_{i}}. Hence Q_{i} acts on L_{i}, so as L_{i}^{∗} = H_{i}^{∗}, L_{i} ⊴ H_{i} and K_{i} = O^{2}(H_{i}) ≤ L_{i}. As |S : Q_{3−i}| = 2^{n}, it follows that

|Q_{i} : C_{Q_{i}}(L_{i})| = |Q_{i} : C_{Q_{i}}(V_{3−i}) ∩ C_{Q_{i}}(V_{3−i}^{h_{i}})| ≤ 2^{2n} = |V_{i} : C_{V_{i}}(K_{i})|,

so Q_{i} = V_{i}C_{Q_{i}}(K_{i}) and C_{Q_{i}}(K_{i}) ≤ C_{Q_{i}}(V_{3−i}). By definition, D_{i} = C_{S}(K_{i}) ≤ Q_{i}, so C_{Q_{i}}(K_{i}) = D_{i}, completing the proof of (2).

By (2), V_{i}D_{i} = Q_{i} = V_{i}C_{S}(V_{1}V_{2}), so

Φ(C_{S}(V_{1}V_{2})) = Φ(V_{i}C_{S}(V_{1}V_{2})) = Φ(V_{i}D_{i}) = Φ(D_{i}),

so (5) holds. By (5), Φ(D_{1}) = Φ(D_{2}) ≤ D_{1} ∩ D_{2} = D_{0}. Set S^{+} = S/Φ(D_{1}). Then Z(S^{+}) = E_{i}^{+}D_{i}^{+} with m(E_{i}^{+}/E_{i}^{+} ∩D_{i}^{+}) = n, so m(D_{i}^{+}/D_{1}^{+} ∩D_{2}^{+}) ≤ m(Z(S^{+})/D_{2}^{+}) = n, establishing (6).

Corollary 1.12. Let τ = (H, F, V) ∈ T (Baum(S)) and set C_{1} = C_{1}(τ, Aut(Baum(S))) and C_{2} = Ω_{1}(Z(S)) ∩ C_{2}(τ, Aut(Baum(S))). Then (C_{1}, C_{2}) is a GN-pair for S.

Proof. Let T = Baum(S) and τ_{2} = (H_{2}, F_{2}, V_{2}) ∈ T (T). By 1.10 one of the following holds:

(i) C_{1} ≤ Z(H_{2}).

(ii) C_{2} ⊴ H_{2}.

(iii) For some α ∈ Aut(T), [V^{α}, V_{2}] ≠ 1.

Now there exists τ_{α} = (H_{α}, F_{α}, V_{α}) ∈ T (S) with F_{α} = F^{α} and V_{α} = V^{α}, so in case (iii), 1.11.1 says that O^{2}(H_{2}) is a block. This completes the proof.

(1.13) Assume S = Baum(S) and (H, F, V) ∈ T (S). Let Inn(S) ≤ Λ ≤ Aut(S), and set C_{1} = ⟨C_{V} (S)^{Λ}⟩ and C_{2} = ⟨F^{Λ}⟩. Then the following are equivalent:

(1) For each α ∈ Λ, [V, V^{α}] = 1.

(2) C_{1} ≤ Z(H) or C_{2} ⊴ H.

(3) C_{2} ⊴ H.

(4) C_{2} is abelian.

Proof. Assume (1). Part (1) implies part (2) by 1.10. As C_{V} (S) ≰ Z(H), C_{1} ≰ Z(H), so (2) implies (3).

Assume (3). Then for α ∈ Λ, F^{α} ≤ C_{2} ≤ O_{2}(H) = C_{S}(V). Further F^{α} ≤ A ∈ A(S) and by 1.4.5, F = V C_{F} (A). But as F^{α} centralizes V, V C_{F} (A) ≤ C_{S}(F^{α}), so [F, F^{α}] = 1. Therefore (3) implies (4).

Finally assume (4). Then ⟨V^{Λ}⟩ ≤ C_{2} ≤ C_{S}(V), so (1) holds.

Section 2. Fusion systems

In this section p is a prime, S is a finite p-group, and F is a saturated fusion system on S.

The definition of a fusion system appears in Definition 1.1 in [BLO], and the definition of a saturated fusion system appears in Definition 1.2 in [BLO]. Roughly speaking, a fusion system on S is a category whose objects are the subgroups of S, and such that the set hom_{F}(P, Q) of morphisms between subgroups P and Q of S is a set of injective group homomorphisms from P into Q satisfying some weak axioms. If G is a finite group and S ∈ Syl_{p}(G), then F_{S}(G) is the fusion system on F such that hom_{F_{S}(G)}(P, Q) consists of the conjugation maps c_{g} : x ↦ x^{g} for g ∈ G with P^{g} ≤ Q. Again, roughly speaking, F is saturated if it satisfies some axioms that are easily verified for F_{S}(G) using Sylow’s Theorem.

The reader is referred to Section 1 and Appendix A of [BLO] for notation, terminology, and basic results about fusion systems. However we also record some of this notation and terminology in this section, plus introducing some new notation.

Let P ≤ S, and let P^{F} = {P ϕ : ϕ ∈ hom_{F}(P, S)} be the set of F-conjugates of P. For Q ∈ P^{F}, R ≤ P, and α ∈ hom_{F}(P, Q), α induces the injection α^{∗} : hom_{F}(R, P) → hom_{F}(Rα, Q) defined by α^{∗} : ϕ ↦ α^{−1}ϕα.

Recall P is fully normalized, centric if for all Q ∈ P^{F}, |N_{S}(P)| ≥ |N_{S}(Q)|, C_{S}(Q) ≤ Q, respectively. Further P is radical if O_{p}(Aut_{F}(P)) = Inn(P). Write F^{f}, F^{c}, F^{r} for the set of nontrivial fully normalized, centric, radical subgroups of S, respectively, and for X ⊆ {f, r, c}, set

F^{X} = ⋂_{x∈X} F^{x}.

Let S = {F_{i} : i ∈ I) be a set of subcategories F_{i} of F. Define the subsystem of F generated by S to be the smallest fusion system contained in F and containing each member of S. Write ⟨S⟩ for this subsystem. Thus ⟨S⟩ is the intersection of all fusion subsystems of F containing each member of S.

Recall that for P ∈ F^{f}, N_{F}(P), C_{F}(P) is the fusion system T on T = N_{S}(P), C_{S}(P) such that for Q ≤ T, hom_{T} (Q, T) consists of those ϕ ∈ hom_{F}(Q, T) such that ϕ extends to ϕ̂ ∈ hom_{F}(P Q, T) normalizing, centralizing P, respectively. By a result of Puig (appearing as Proposition A.6 in [BLO]), T is a saturated fusion system on T.

We say P is normal in F if F = N_{F}(P). Further F is constrained if F has a normal centric subgroup. If F is constrained then by Proposition C in [BCGLO], there is a finite group G with S ∈ Syl_{p}(G), F^{∗}(G) = O_{p}(G), and F = F_{S}(G). Moreover G is unique up to an isomorphism extending the identity map on S, and we refer to any such group as a model for F. In particular if U ∈ F^{f} with N_{F}(U) constrained, then there exists a model for N_{F}(U), which we denote by G(U). Note that if U ∈ F^{fc} then N_{F}(U) is constrained. Write 𝔊(F) for the set of models G(U), as U varies over the members of F^{f} with N_{F}(U) constrained. For G ∈ 𝔊(F), write S(G) for the Sylow group S ∩ G of G.

Write 𝔄(P) for the set of α ∈ hom_{F}(N_{S}(P), S) such that P α ∈ F^{f}. By A.2.b in [BLO]:

(2.1) For each P ≤ S and Q ∈ F^{f} ∩ P^{F}, there exists α ∈ 𝔄(P) with P α = Q.

(2.2) Let P, Q ≤ S with [P, Q] ≤ P ∩ Q, P ∈ F^{f}, and Q ∈ N_{F}(P)^{f}. Then for each α ∈ 𝔄(Q):

(1) P α ∈ N_{F}(Qα)^{f} and (N_{S}(P) ∩ N_{S}(Q))α = N_{S}(P α) ∩ N_{S}(Qα).

(2) α : N_{N_{F} (P)}(Q) → N_{N_{F} (Qα)}(P α) is an isomorphism.

(3) If N_{F}(P) and N_{F}(Qα) are constrained then α : N_{S}(P) ∩ N_{S}(Q) → N_{S}(P α) ∩ N_{S}(Qα) extends to an isomorphism α̌ : N_{G(P)}(Q) → N_{G(Qα)}(P α).

(4) If Q ∈ F^{f} then N_{N_{F} (P)}(Q) = N_{N_{F} (Q)}(P). Further if N_{F}(P) and N_{F}(Q) are constrained, then the identity map ι on N_{S}(P) ∩ N_{S}(Q) extends to an isomorphism ι̌: N_{G(P)}(Q) → N_{G(Q)}(P).

(5) If Q ∈ F^{f}, Q ⊴ N_{F}(P), and N_{F}(P) and N_{F}(Q) are constrained, then the identity map ι on N_{S}(P) extends to an isomorphism ι̌: G(P) → N_{G(Q)}(P).

Proof. Let T = N_{S}(P), Q̄ = Qα, and S̄ = N_{S}(Q̄). More generally for R ≤ N_{T} (Q) set R̄ = Rα. Then N_{T} (Q)α ≤ NS̄(P̄), and as P ∈ F^{f}, there exists β ∈ 𝔄(P̄) with P αβ = P̄β = P. Then NS̄(P̄)β ≤ N_{T} (Q̄β) = N_{T} (Qαβ), with αβ acting on P. Thus αβ ∈ N_{F}(P), and as Q ∈ N_{F}(P)^{f} we may choose β so that Qαβ = Q. Then |N_{T} (Q)| ≤ |NS̄(P̄)| ≤ |N_{T} (Q)|, so N_{T} (Q)α = NS̄(P̄). Since we could have chosen α with P̄ ∈ N_{F}(Q̄)^{f}, part (1) follows.

Set N = N_{N_{F} (P)}(Q) and N̄ = N_{N_{F} (Q̄)}(P̄). Let R ≤ N_{T} (Q) and ϕ ∈ hom_{N} (R, T). Then ϕ extends to ϕ̂ ∈ hom_{N} (RP Q, T) acting on P and Q, so ϕ̂α^{∗} extends ϕα^{∗} to R̄P̄Q̄ and acts on P̄ and Q̄, and hence ϕα^{∗} ∈ N̄.

Conversely if φ ∈ homN̄(R̄, T̄) then φ extends to φ̂ ∈ homN̄ (R̄P̄Q̄, T̄) acting on P̄ and Q̄, and then φ̂α^{−∗} extends φα^{−∗}, so φα^{−∗} ∈ hom_{N} (RP Q, T). Therefore α : N → N̄ is an isomorphism, establishing (2). Then (2) and 2.3 and 2.4.1 in [A5] imply (3).

Suppose that Q ∈ F^{f} and let i be the identity map on N_{S}(Q); then i ∈ 𝔄(Q). Then (4) follows from (2) and (3) applied to i in the role of α. Part (5) is a special case of (4). Definition 2.3. Let Ω be a F-invariant collection of subgroups of S; that is all Fconjugates of members of Ω are contained in Ω. Write F_{Ω}^{f} for the set of U ∈ F^{f} such that N_{S}(U) contains a member of Ω, and set F_{Ω}^{frc} = F_{Ω}^{f} ∩ F^{frc}.

A collection Ω′ of subgroups of S is Baumann-closed if Ω′ is F-invariant, and for each U ∈ F_{Ω′}^{f}, Baum(N_{S}(U)) ∈ Ω′.

Define F to be Ω-Alperin if F = ⟨Aut_{F}(U) : U ∈ F_{Ω}^{frc}⟩. Define F to be of Ωcharacteristic p if for each U ∈ F_{Ω}^{f}, N_{F}(U) is constrained. Define F to be an Ω-system if F is both Ω-Alperin and of Ω-characteristic p. For example the systems of characteristic p-type are the systems of ∅-characteristic p, and by Alperin’s Fusion Theorem, such systems are ∅-systems.

For P ≤ S with P^{F} = {P }, F is P -Alperin, P -characteristic p if F is {P }-Alperin, {P }-characteristic p, respectively. Define F to be of Baumann characteristic p if F is of Baum(S)-characteristic p. Definition 2.4. Assume p = 2 and define 𝔐(F) to consist of the minimal parabolics M (cf. Definition 1.1) in some G ∈ 𝔊(F) over S(G) such that:

(1) Either M is solvable, or for L ∈ C(M), L/O_{∞}(L) ≅ L_{2}(2^{e}) or A_{2^{e}+1} for some integer e ≥ 2 depending on M.

(2) Neither Ω_{1}(Z(S(G))) nor Baum(S(G)) are normal in M.

Set S(M) = S(G). Write 𝔐_{−}(F) for the collection of those M ∈ 𝔐(F) such that:

(3) S(M) ≠ S and for each 1 ≠ W ≤ S(G) with W ⊴ M, W ∈ F^{f} and S(M) = N_{S}(W).

Write 𝔐_{+}(F) for the collection of those M ∈ 𝔐(F) such that S(M) = S.

Recall from Definition A.3.1 in [ASm] that C(M) consists of the minimal perfect subnormal subgroups of M.

If Ω is an F-invariant collection of subgroups of S, define 𝔐_{Ω,−}(F) to consist of those M ∈ 𝔐_{−}(F) such that S(M) contains a member of Ω.

(2.5) Assume F is constrained. Then (1) F is of characteristic p-type.

(2) If p = 2 then 𝔐_{−}(F) is empty.

Proof. Let U ∈ F^{f} and suppose N = N_{F}(U) is not constrained. Then by Theorem 6 in [A6], E(N) ≠ 1. But by Theorem 7 in [A6], E(N) ≤ E(F), so E(F) ≠ 1, so F is not constrained by Theorem 6 in [A6], proving (1).

Assume p = 2 and M ∈ 𝔐_{−}(F). Set Q = O_{2}(F), W = O_{2}(M), and T = S(M). Then N_{Q}(W) ≤ N_{S}(W) = T, and then N_{Q}(W) ≤ W. Hence Q ≤ W, impossible as Q ⊴ M but T ≠ S = N_{S}(Q). Thus (2) holds.

(2.6) Assume for each U ∈ F^{f} of order p that N_{F}(U) is constrained. Then F is of characteristic p-type.

Proof. Assume P ∈ F^{f} with N = N_{F}(P) not constrained. Then E(N) ≠ 1 by Theorem 6 in [A6]. Let V be of order p in Z(N_{S}(P))∩O_{p}(N). By 10.3 in [A6], E(N) = E(N_{N} (V)).

Let α ∈ 𝔄(V), U = V α, and M = N_{F}(U). By 2.2, P α ∈ M^{f} and α : N_{N} (V) → N_{M}(P α) is an isomorphism, so 1 ≠ E(N)α = E(N_{M}(P α)), and hence N_{M}(P α) is not constrained by Theorem 6 in [A6]. As M is constrained by hypothesis, this contradicts 2.5.1. Section 3. FT I-subgroups

In this section we assume: Hypothesis 3.1. F is a saturated fusion system on a finite p-group S and U is a nontrivial abelian subgroup of S such that U ∈ F^{f}.

Recall that if G is a group then a subgroup T of G is a TI-subgroup of G if for each g ∈ G − N_{G}(T), T ∩ T^{g} = 1. The object of this section is to translate the notion of an abelian TI-subgroup into the domain of fusion systems.

Recall from section 2 that for P ≤ S, 𝔄(P) denotes the set of α ∈ hom_{F}(N_{S}(P), S) such that P α ∈ F^{f}.

Definition 3.2. The fully normalized abelian subgroup U of S is an FT I-subgroup of S if whenever P ≤ N_{S}(U) and ϕ ∈ hom_{F}(P, S) with 1 ≠ (U ∩ P)ϕ ∩ U, then ϕ extends to ϕ̂ ∈ hom_{F}(UP, S) with Uϕ̂ = U.

(3.3) Assume U is an FT I-subgroup of S, 1 ≠ V ≤ U, and α ∈ 𝔄(V). Then (1) N_{S}(V α) ≤ N_{S}(Uα), and

(2) Uα ⊴ N_{F}(V α).

Proof. Let V̄ = V α, Ū = Uα, and S̄ = N_{S}(V̄). Suppose that S̄ ≠ NS̄(Ū), and pick t ∈ NS̄(NS̄(Ū)) with Ū^{t} ≠ Ū. By 2.1 there is β ∈ 𝔄(Ū) with Ūβ = U. Then c_{t}β^{∗} ∈ hom_{F}(U, N_{S}(U)) with 1 ≠ V̄ β ≤ U ∩ U(c_{t}β^{∗}), so as U is FT I, U = U(c_{t}β^{∗}). Therefore Ū = Ū^{t}, a contradiction. So (1) is established.

Let P ≤ S̄ and ϕ ∈ hom_{N_{F} (V̄)}(P, S̄). Then ϕ extends to φ ∈ hom_{F}(V̄ P, S̄) acting on V̄. Now φβ^{∗} ∈ hom_{F}((V̄ P)β, S) with φβ^{∗} acting on V̄ β ≤ U, so as U is FT I, it follows that φβ^{∗} extends to γ ∈ hom_{F}(UP β, S) acting on U. Then γβ^{−∗} extends ϕ to ŪP, and acts on Ū. Therefore Ū ⊴ N_{F}(V̄), establishing (2).

(3.4) U is an FT I-subgroup of S iff the following two conditions are satisfied: (1) For all V ≤ U and ϕ ∈ hom_{F}(V, U), ϕ extends to a member of Aut_{F}(U).

(2) For each x ∈ U^{#} and α ∈ 𝔄(x), Uα ⊴ C_{F}(xα).

Proof. First assume that U is FT I. Then if 1 ≠ V ≤ U and ϕ ∈ hom_{F}(V, U) then 1 ≠ V ϕ = V ϕ ∩ U, so condition (1) is satisfied by the FT I-condition. Further condition (2) holds by 3.3.2.

Conversely assume that conditions (1) and (2) are satisfied. Suppose P ≤ N_{S}(U) and ϕ ∈ hom_{F}(P, S) with U ∩ (U ∩ P)ϕ ≠ 1. Set V = U ∩ P. As P ≤ N_{S}(U), V ⊴ P, so V ϕ ⊴ P ϕ.

Pick x ∈ V ϕ∩U^{#}. Then xϕ^{−1} ∈ x^{F} ∩U, so by condition (1), there exists φ ∈ Aut_{F}(U) with xϕ^{−1}φ = x. Then β = ϕ^{−1}φ ∈ hom_{F}(V ϕ, U), and β fixes x. Let α ∈ 𝔄(x). By condition (2), Uα ⊴ C_{F}(xα), so βα^{∗} extends to γ ∈ hom_{F}(UαV ϕα, S) fixing xα and Uα. As V ϕβ ≤ U, (V ϕα)(βα^{∗}) ≤ Uα, so (UαV ϕα)γ = Uα. Therefore V ϕ ≤ U.

As V ϕ ⊴ P ϕ and V ϕ ≤ U, we may pick x ∈ Z(P ϕ). Therefore P ϕ ≤ C_{S}(x), so P ϕα ≤ C_{S}(xα) ≤ N_{S}(Uα). Thus P ϕ ≤ N_{S}(U). Then as U ∈ F^{f} and U ⊴ N_{F}(U), ϕ extends to a member of hom_{F}(P U, S) acting on U.

(3.5) If U is an FT I-subgroup of S then for each V ≤ U there is α ∈ 𝔄(V) with V α ≤ U.

Proof. We may assume V ≠ 1 and pick α ∈ 𝔄(V). By 3.3.1, N_{S}(V α) ≤ N_{S}(Uα). Let β ∈ 𝔄(Uα) with Uαβ = U, and set γ = αβ. Then V γ ≤ U and N_{S}(V α)β ≤ N_{S}(Uα)β ≤ N_{S}(U) ≤ S, so as V α ∈ F^{f}, also V γ = (V α)β ∈ F^{f}. Definition 3.6. Assume U is an abelian FT I-subgroup of S. Define

P = P(U) = {1 ≠ P ≤ S : hom_{F}(P, U) ≠ ∅}.

For P ∈ P let 𝔅(P) = 𝔅(P, U) consist of those β ∈ hom_{F}(Q, S) such that P ≤ Q ≤ N_{S}(P) and P β ≤ U. Write Q_{β} for the domain Q of β, and set U_{β} = U ∩ Qβ. Set U(P) = {U_{β}β^{−1} : β ∈ 𝔅(P)}.

(3.6) Assume U is an FT I-subgroup of S. Then P(U) is F-invariant.

Proof. Trivial.

(3.7) Assume U is an FT I-subgroup of S and let P ∈ P(U). Then (1) There is a largest member U(P) of U(P).

(2) For each α_{1} ∈ 𝔄(P) with P α_{1} ≤ U, U(P) = U_{α_{1}}α_{1}^{−1}.

(3) For each β ∈ 𝔅(P), U_{β}β^{−1} = U(P) ∩ Q_{β}.

Proof. Let α_{1}, β ∈ 𝔅(P) such that P α_{1} = P β = V_{1} ∈ F^{f}, V_{1} ≤ U, and Q_{β} ≤ Q_{α_{1}}. Then γ = β^{−1}α_{1} ∈ hom_{N_{F} (V_{1})}(Q_{β}β, Q_{β}α_{1}), so by 3.3.2, γ extends to γ̂ ∈ hom_{F}(UQ_{β}β, UQ_{β}α_{1}) acting on U. Hence

U_{β}β^{−1}α_{1} = (U ∩ Q_{β}β)β^{−1}α_{1} = (U ∩ Q_{β}β)γ̂ = Uγ̂ ∩ Q_{β}βγ̂ = U ∩ Q_{β}α_{1} = Q_{β}α_{1} ∩ U ∩ Q_{α_{1}}α_{1} = Q_{β}α_{1} ∩ U_{α_{1}},

so U_{β}β^{−1} = Q_{β} ∩ U_{α_{1}}α_{1}^{−1} ≤ U_{α_{1}}α_{1}^{−1}.

In particular given α_{1} ∈ 𝔄(P) with P α_{1} = V_{1} ≤ U, and β ∈ 𝔅(P) with P β = V_{1}, we have Q_{β} ≤ N_{S}(P) = Q_{α_{1}}, so

(*)  U_{β}β^{−1} = Q_{β} ∩ U_{α_{1}}α_{1}^{−1} ≤ U_{α_{1}}α_{1}^{−1}.

Now let α ∈ 𝔅(P) and set V = P α. Then there exists ζ ∈ 𝔄(V) with V ζ = V_{1}. Claim Uζ = U. Namely 1 ≠ V_{1} = U ∩ V_{1} = U ∩ V ζ, so as U is an FT I-subgroup of S, ζ_{|V} extends to γ ∈ Aut_{F}(U). Then ζ^{−1}γ ∈ hom_{F}(Uζ, U) acts on V_{1}, so as U ⊴ N_{F}(V_{1}) by 3.3.2, indeed Uζ = U.

Next β = αζ ∈ 𝔅(P) with Q_{β} = Q_{α} and P β = V_{1}, so by paragraph one of this proof,

(**)  U_{β}β^{−1} = Q_{β} ∩ U_{α_{1}}α_{1}^{−1} ≤ U_{α_{1}}α_{1}^{−1}.

Also

U_{β}β^{−1} = (U ∩ Q_{β}β)β^{−1} = (U ∩ Q_{α}αζ)ζ^{−1}α^{−1} = (U ∩ Q_{α}α)α^{−1} = U_{α}α^{−1},

so by (*) and (**),

(!)  U_{α}α^{−1} = U_{β}β^{−1} = Q_{β} ∩ U_{α_{1}}α_{1}^{−1} = Q_{α} ∩ U_{α_{1}}α_{1}^{−1} ≤ U_{α_{1}}α_{1}^{−1},

establishing (1) and (2). Indeed from (!), U_{α}α^{−1} = Q_{α} ∩ U_{α_{1}}α_{1}^{−1} = Q_{α} ∩ U(P), so (3) holds too.

(3.8) Assume U is an FT I-subgroup of S and let P ∈ P(U). Then (1) N_{S}(P) ≤ N_{S}(U(P)).

(2) P ≤ U(P) ≤ N_{S}(P).

(3) If P_{1} ∈ P with P_{1} ≤ P then U(P_{1}) = U(P).

(4) U(U(P)) = U(P).

(5) If P′ ∈ P with U(P) ∩ U(P′) ≠ 1 then U(P) = U(P′).

(6) For ϕ ∈ hom_{F}(U(P), S), U(P)ϕ ≤ U(P ϕ).

(7) For s ∈ S, U(P)^{s} = U(P^{s}).

(8) U(P) is a TI-subgroup of S.

Proof. Let α ∈ 𝔄(P) with P α ≤ U. By 3.7, U(P) = U_{α}α^{−1}. Now Q_{α} = N_{S}(P) and U_{α} = U ∩ Q_{α}α ⊴ Q_{α}α as U ⊴ N_{F}(P α) by 3.3.2. Thus U(P) = U_{α}α^{−1} ⊴ Q_{α}, establishing (1).

Next P α ≤ N_{S}(P)α ∩ U = U_{α}, so P ≤ U_{α}α^{−1} = U(P), and as U is abelian, so is U(P) so U(P) ≤ N_{S}(P). Thus (2) holds.

Let P_{1} ∈ P with P_{1} ≤ P. As U is abelian, P_{1} ≤ P ≤ U(P) ≤ N_{S}(P_{1}). Thus for β ∈ 𝔅(U(P)), we have β ∈ 𝔅(P_{1}) with U(P)β ≤ U_{β}, so U(P) ≤ U(P_{1}). Let γ ∈ 𝔅(U(P_{1})). Then γ ∈ 𝔅(P) with U(P_{1})γ ≤ U_{γ}, so U(P_{1}) ≤ U(P), establishing (3).

As P ≤ U(P), U(P) = U(U(P)) by (3), proving (4).

Suppose P′ ∈ P with P̄ = U(P) ∩ U(P′) ≠ 1. Then P̄ ∈ P with P̄ contained in U(P) and U(P′), so by (3) and (4),

U(P) = U(U(P)) = U(P̄) = U(U(P′)) = U(P′),

proving (5).

Let ϕ ∈ hom_{F}(U(P), S). For α ∈ 𝔄(P) with P α ≤ U, β = ϕ^{−1}α ∈ hom(U(P)ϕ, U) is in 𝔅(P ϕ), so U(P)ϕ ≤ U(P ϕ), establishing (6). Then for s ∈ S, U(P)^{s} ≤ U(P^{s}) by (6), and similarly as (P^{s})^{s^{−1}} = P, U(P^{s})^{s^{−1}} ≤ U(P), completing the proof of (7). Then (5) and (7) imply (8).

Section 4. Classical groups

In this section we assume:

Hypothesis 4.1. q = 2^{e} for some positive integer e, and either L is a simple classical group over the field F_{q} or L is G_{2}(q). Set A = Aut(L), regard L as the subgroup Inn(L) of A, and set Ā = A/L = Out(L). Pick Σ ∈ Syl_{2}(Ā) and let S ≤ A with S ∩ L = 1 and S̄ ≤ Σ. Let Φ ≤ Σ be a Sylow 2-subgroup of the group of field automorphisms of L.

(4.2) Assume Hypothesis 4.1. Then (1) For each s ∈ S with s̄ ∈ Φ, s induces a field automorphism on L.

(2) If L ≅ Sp_{n}(q) with n ≠ 4 then Σ = Φ.

(3) If L ≅ Sp_{4}(q) then Σ is cyclic, |Σ : Φ| = 2, and if s ∈ S with s̄ ∈ Σ − Φ then C_{L}(s) ≅ Sz(2^{m}), where e = 2^{a}m with m odd.

(4) If L ≅ L_{n}(q) with n > 2 then Σ = ⟨τ⟩ × Φ with τ a graph automorphism of order 2. Further

(a) If s ∈ S with s̄ = τ then either:

(i) n is odd and C_{L}(s) ≅ Sp_{n−1}(q), or (ii) n is even and either O_{2}(C_{L}(s)) ≠ 1 or C_{L}(s) ≅ Sp_{n}(q).

(b) If s ∈ S with s̄ = τϕ with ϕ ∈ Φ an involution, then C_{L}(s) ≅ U_{n}(q^{1/2}).

(5) If L ≅ U_{n}(q) with n > 2 then Σ is cyclic, and if s ∈ S is an involution then either:

(i) n is odd and C_{L}(s) ≅ Sp_{n−1}(q), or (ii) n is even and either O_{2}(C_{L}(s)) ≠ 1 or C_{L}(s) ≅ Sp_{n}(q).

(6) If L ≅ Ω_{n}^{−}(q) with n ≥ 8, then Σ is cyclic, and if s is an involution in S then either O_{2}(C_{L}(s)) ≠ 1 or C_{L}(s) ≅ Sp_{n−2}(q).

(7) If L ≅ Ω_{n}^{+}(q) with n ≥ 8, then Σ = ⟨τ⟩ × Φ with τ a graph automorphism of order 2. Further

(a) If s ∈ S with s̄ = τ then ⟨s⟩L ≅ O_{n}^{+}(2) and either O_{2}(C_{L}(s)) ≠ 1 or C_{L}(s) ≅ Sp_{n−2}(q).

(b) If s ∈ S with s̄ = τϕ and ϕ an involution in Φ, then C_{L}(s) ≅ Ω_{n}^{−}(q^{1/2}).

(8) If L ≅ G_{2}(q) then Σ = Φ, and if s ∈ S is an involution then C_{L}(s) ≅ G_{2}(q^{1/2}).

Proof. The structure of Σ can be obtained from Theorem 2.5.12 in [GLS3]. The classes of involutions and centralizers appear in section 19 of [ASe].

(4.3) Assume Hypothesis 4.1. Assume also that T is a 2-subgroup of A such that N_{L}(T) has a strongly embedded subgroup. Then one of the following holds:

(1) L ≅ L_{2}(q) or U_{3}(q).

(2) L ≅ L_{3}(q), Sp_{4}(q), or A_{6} ≅ Sp_{4}(2)′, and T is nontrivial on the Dynkin diagram of L.

Proof. The proof is by induction on the order of L, so we may assume L, T is a counter example to the lemma with |L| mininimal. Set K = N_{L}(T).

As K has a strongly embedded subgroup:

(a) O_{2}(K) = 1 and either m_{2}(K) = 1 or O^{2′}(K/O(K)) is a group of Lie type of even characteristic and Lie rank 1.

By (a), O_{2}(K) = 1, so T ∩ L = 1. Thus:

(b) K = C_{L}(T) and we may take T = S to satisfy the hypothesis of 4.1. If S = 1 then conclusion (1) of the lemma holds by (a), contrary to the choice of

L, T as a counter example. Thus S ≠ 1, so we can pick an involution z ∈ Z(S). Set H = C_{L}(z). Then

(c) K = C_{H}(S). In particular as O_{2}(K) = 1, it follows from (c) that:

(d) O_{2}(H) = 1. We conclude from (d) and 4.2 that either H is a classical group of even characteristic,

or L ≅ G_{2}(q) and H ≅ G_{2}(q^{1/2}), or L ≅ Sp_{4}(q) with e odd, and z is nontrivial on the Dynkin diagram of L. However in the third case, 4.2 says that S = ⟨z⟩, so that conclusion (2) of the lemma holds, contrary to the choice of L, T as a counter example. Therefore we have shown:

(e) H is a classical group of even charactersitic, or L ≅ G_{2}(q) and H ≅ G_{2}(q^{1/2}). By (c), (e), and the minimality of |L|, H satisfies one of the conclusions of the lemma,

so:

(f) For some even prime power q_{0}, one of the following holds:

(i) H ≅ L_{2}(q_{0}) or U_{3}(q_{0}), or (ii) H ≅ L_{3}(q_{0}) or Sp_{4}(q_{0}), and S is nontrivial on the diagram of H. We now inspect 4.2 for the possibilities for z, in light of the structure of H described

in (f). If z is a field automorphism then L ≅ L_{2}(q) in (fi), while in (fii), L ≅ L_{3}(q) or Sp_{4}(q), and as S is nontrivial on the diagram of H, S is also nontrivial on the diagram of L. Hence the lemma holds in this case, contrary to the choice of L, T.

Next if z̄ = τ is a graph automorphism, or L ≅ U_{n}(q), then by (d) and 4.2, L ≅ L_{n}^{ϵ}(q) with n odd, and H ≅ Sp_{n−1}(q); or L ≅ L_{n}^{ϵ}(q) with n even and H ≅ Sp_{n}(q); or L ≅ Ω_{n}^{+}(q) with n ≥ 8, and H ≅ Sp_{n−2}(q). We conclude from (f) that L ≅ L_{4}^{ϵ}(q) or L_{5}^{ϵ}(q), and S is nontrivial on the diagram of H. This is impossible as from 4.2, S induces a group of field automorphisms on H, and therefore is trivial on the diagram of H.

It follows from 4.2 that:

(g) z induces a graph-field automorphism on L, and L is not a unitary group.

If L ≅ L_{n}(q) then by (d), (g) and 4.2, H ≅ U_{n}(q^{1/2}). Then by (f), n = 3. Further as z is a graph-field automorphism, z is nontrivial on the diagram of L, so conclusion (2) of the lemma holds, contrary to the choice of L, T.

We have reduced to the case where L ≅ Ω_{n}^{ϵ}(q) with n ≥ 8, and z induces a graph-field automorphism on L. Then by (d) and 4.2, H ≅ Sp_{n−2}(q) or Ω_{n}^{−}(q^{1/2}), contrary to (f). The proof is at last complete.

Section 5. 2-Fusion in symmetric and alternating groups

In this section we assume:

Hypothesis 5.1. Ω = {1, …, n} is a finite set, M = Sym(Ω), L = Alt(Ω), G is L or M, S_{M} ∈ Syl_{2}(M), and S = S_{M} ∩ G ∈ Syl_{2}(G). Let T be the set of transpositions in S_{M} and T_{1} = ⟨T ⟩ ∩ S. Let P = P_{1}(S_{M}, Ω) be the set of orbits of ⟨T ⟩ on Ω, regarded as a partition of Ω.

Throughout this section we adopt the following notation involving permutation groups. A [1^{m_{1}}, …, n^{m_{n}}]-partition of Ω is a partition with m_{i} blocks of size i. A regular (m, k)- partition is a partition with k blocks each of size m. For X ⊆ M, write F ix(X), Mov(X) for the set of points of Ω fixed, moved by X, respectively. For Γ ⊆ Ω, let X_{Γ} be the pointwise stabilizer in X of Γ, and N_{X}(Γ) the global stabilizer in X of Γ. If X acts on Δ ⊆ Ω write X^{Δ} for the image of X in Sym(Δ) under the restriction map.

Define a partial ordering on the set of partitions of Ω by P ≤ Q if Q is a refinement of P.

Given a group Y with Sylow 2-subgroup T, write B(Y, T) for the set of subgroups B of T such that N_{T} (B) ∈ Syl_{2}(N_{Y} (B)) and B = O_{2}(N_{Y} (B)). Write B_{∗}(Y, T) for those B ∈ B(Y, T) such that N_{Y} (B)/B has a strongly embedded subgroup.

(5.2) (1) If n is even then P is a regular (2, n/2)-partition.

(2) If n is odd then P is a [2^{n−1}/2, 1]-partition.

(3) If (n, G) ≠ (4, L) then T_{1} is weakly closed in S with respect to G. If (n, G) ≠ (2, L) then P is the set of orbits of T_{1} on Ω.

(4) If (n, G) ≠ (4, L) then N_{G}(S) ≤ N_{G}(T) ≤ N_{G}(P).

(5) Either

(i) S = N_{G}(S), or

(ii) 3 ≤ n ≤ 5, G = L, and |N_{G}(S) : S| = 3.

Proof. If s, t ∈ T are distinct, then as ⟨s, t⟩ is a 2-group, Mov(s) ∩ Mov(t) = ∅. On the other hand if T′ is a set of transpositions with this property of maximal order, then |T′| = [n/2], and the set P′ of orbits of ⟨T′⟩ on Ω is a partition of Ω of the type described in (1) or (2). Then (1) and (2) follow from Sylow’s Theorem.

The first statement in (3) is 4.4.4 in [ASh]. By inspection, if (n, G) ≠ (2, L) then P is the set of orbits of T_{1} on Ω. Thus (3) holds, while (3) implies (4).

We prove (5) by induction on n. When n ≤ 5, the result follows by inspection, so we may assume n > 5. By (4), N_{G}(S) ≤ N_{G}(P). Let Q be the set of blocks of P of size 2, so that Q = P if n is even, and Q = P − {α} for {α} = F ix(S) if n is odd. Then also N_{G}(S) ≤ N_{G}(Q), and T_{1} = G_{Q}. Now N_{G}(Q)^{Q} = Sym(Q) and N_{G}(S)^{Q} = N_{Sym(Q)}(S^{Q}) = S^{Q} by induction on n. This completes the proof of (5).

Notation 5.3. Let

n = ∑_{i=0}^{∞} a_{i}2^{i}

with a_{i} ∈ {0, 1} be the binary expansion of n, and let

I = {i ∈ I : a_{i} = 1}.

Let P(n) be the set of partitions {Ω_{i} : i ∈ I} of Ω such that |Ω_{i}| = 2^{i} for i ∈ I. Write P (S_{M}) for the set of orbits of S_{M} on Ω, regarded as a partition of Ω.

Set a(n) = max{i : a_{i} = 1}.

(5.4) (1) P (S_{M}) = (Ω_{i} : i ∈ I) ∈ P(n).

(2) We have

S_{M} = ∏_{i∈I} S_{M}(i)

is the direct product of the groups S_{M}(i) = (S_{M})_{Ω−Ω_{i}} for i ∈ I.

(3) P (S_{M}) ≤ P.

(4) We have

Z(S_{M}) = ∏_{0≠i∈I} Z_{M}(i),

where Z_{M}(i) = ⟨z(i)⟩ and z(i) is the involution whose cycles of length 2 are the blocks of P contained in Ω_{i}.

(5) If (n, G) ≠ (2, L) or (3, L) then P (S_{M}) is the set P (S) of orbits of S on Ω.

Proof. Parts (1)-(4) appear in 7.3 in [A2]. If i > 1 then as S_{M}(i) ∈ Syl_{2}(M_{Ω−Ω_{i}}), S_{M}(i) contains an abelian subgroup regular on Ω_{i}, so Ω_{i} is an orbit of S. If i = 1 and n > 3 then there is 1 < j ∈ I and S_{M} ∩ L contains an involution s moving 4 points of Ω with a cycle of length 2 on Ω_{1} and a cycle of length 2 on Ω_{j}, so Ω_{1} is an orbit of S. Therefore (5) holds.

Definition 5.5. We define a sequence of partitions and subsets of Ω, using a recursive definition.

In 5.1, we defined P_{1} = P_{1}(S_{M}, Ω). If n is even, set P̄_{1} = P̄_{1}(S_{M}, Ω) = P_{1}, P′_{1} = P′_{1}(S_{M}, Ω) = P_{1}, Δ_{1} = Δ_{1}(S_{M}, Ω) = Ω, and Δ′_{1} = Δ′_{1}(S_{M}, Ω) = ∅. If n is odd then P contains a unique block {α} of size 1, and we set P̄_{1} = P −{α}, P′_{1} = {α}, Δ_{1} = Ω−{α}, and Δ′_{1} = {α}. We regard P̄_{1} as a partition of Δ_{1} and P′_{1} as a partition of Δ′_{1}. Observe that Δ′_{1} is a complement to Δ_{1} in Ω. Further when n is odd, from Notation 5.3, Ω_{0} = {α}, so that

P_{1} = P̄_{1} ∪ {Ω_{0}} and Δ_{1} = Ω − Ω_{0} = ⋃_{i>0} Ω_{i} when n is odd.

Now assume P_{k} = P_{k}(S_{M}, Ω), P̄_{k}, etc. are defined. Proceeding recursively, we define P_{k+1} = P_{k+1}(S_{M}, Ω), P̄_{k+1}, etc. First let Q_{k+1} = P_{1}(Sym(P̄_{k}), SP̄_{k}). Then define P̃_{k+1} to be the partition of Δ_{k} induced by Q_{k+1}; that is for A ∈ Q_{k+1}, let Ã be the union of the blocks of P̄_{k} contained in A, and set P̃_{k+1} = {Ã : A ∈ Q_{k+1}}. If |P̄_{k}| is even, set P̄_{k+1} = P̃_{k+1}, Δ_{k+1} = Δ_{k}, P′_{k+1} = P′_{k}, and Δ′_{k+1} = Δ′_{k}. If |P̄_{k}| is odd, let A be the block of Q_{k+1} of size 1, P̄_{k+1} = P̃_{k+1} − {Ã}, Δ_{k+1} = Δ_{k} − Ã, P′_{k+1} = P′_{k} ∪ {Ã}, and Δ′_{k+1} = Δ′_{k} ∪ Ã. In any event, define P_{k+1} to be P̄_{k+1} ∪ P′_{k+1}, regarded as a partition of Ω.

Set P_{i}^{∘} = P_{i}^{∘}(S_{M}, Ω) = P̄_{i} ∪ {Δ′_{i}}, regarded as a partition of Ω. Define P_{0} to be the trivial partition ∞ of order n of Ω.

Given partitions Q, R of Ω with R ≤ Q, we can form the partition R/Q on Q induced by R; that is for D ∈ R let D^{+} be the set of blocks of Q contained in D, and set R/Q = {D^{+} : D ∈ R}.

(5.6) Let a = a(n). Then for each 1 ≤ k ≤ a:

(1) Δ′_{k} is a complement to Δ_{k} in Ω.

(2) P_{k}, P̄_{k}, P′_{k}, and P_{k}^{∘} are S-invariant partitions of Ω, Δ_{k}, Δ′_{k}, and Ω, respectively.

(3) For k < a, P_{k+1} < P_{k}.

(4) The blocks of P̄_{k} are of size 2^{k}.

(5) We have

P′_{k} = {Ω_{i} : i < k} and Δ′_{k} = ⋃_{i<k} Ω_{i}.

(6) P (S_{M}) = P_{a} ≤ P_{k}.

(7) Set T_{k} = S_{P_{k}}. If (n, G) ≠ (4, L) then T_{k} is weakly closed in S with respect to G.

(8) If (n, G) ≠ (2, L) then P_{k} is the set of orbits of T_{k} on Ω.

(9) If (n, G) ≠ (3, L), (4, L), or (5, L), then P_{k} is the unique S-invariant partition of its type.

Proof. Parts (1)-(4) follow by induction on k, using the recursive construction in Definition 5.5.

We next prove (5) by induction on k. When k = 1, (5) is a consequence of parts (1) and (2) of 5.2, and the construction of P′_{1} in 5.5. Thus we may assume the result at some k ≥ 1, and it remains to prove (5) at k + 1. By hypothesis P′_{k} = {Ω_{i} : i < k}, and by (4) the blocks of P̄_{k} are of size 2^{k}. It follows that |P̄_{k}| is even iff |Δ_{k}| ≡ 0 mod 2^{k+1}. By (2), P′_{k} is a partition of Δ_{k}, while by 5.4.1, {Ω_{i} : i ∈ I} is a partition of Ω, so {Ω_{i} : i ≥ k} is a partition of Δ_{k}. Then as |Ω_{i}| = 2^{i} for i ∈ I, it follows that |Δ_{k}| ≡ 0 mod 2^{k+1} iff a_{k} = 0. Moreover in that event

⋃_{i<k} Ω_{i} = ⋃_{i<k+1} Ω_{i},

while from Definition 5.5, Δ′_{k+1} = Δ′_{k}, so (5) holds in this case. So suppose a_{k} = 1, or equivalently that |P̄_{k}| is odd. Then from Definition 5.5, Ω_{k} is the fixed block of S_{M} on P̃_{k+1}, so P′_{k+1} = P′_{k} ∪ {Ω_{k}}, and

Δ′_{k+1} = Δ′_{k} ∪ Ω_{k} = ⋃_{i<k+1} Ω_{i},

completing the proof of (5).

By (5), P′_{a} = {Ω_{i} : i < a}, and by (1), Δ′_{a} is a complement to Δ_{a} in Ω, so Δ_{a} = Ω_{a}. By (4), the blocks of P̄_{k} are of size 2^{k}, and by (2), P̄_{a} is a partition of Δ_{a}, so P̄_{a} = {Ω_{a}}. Thus P (S_{M}) = P_{a}. By (3), P_{a} ≤ P_{k}, completing the proof of (6).

We next prove (7) and (8) by induction on k. When k = 1, (7) and (8) follow from 5.2.3. Thus we may assume (7) and (8) hold at some k ≥ 1, and it remains to show they also hold at k + 1. By construction in Definition 5.5, T_{k+1}/T_{k} is the subgroup of SP̄_{k} generated by its transpositions. Hence by 5.2.3, P̃_{k+1} is the set of orbits of T_{k+1} on Δ_{k}, and as (8) holds at k, P′_{k} is the set of orbits of T_{k}, and hence also T_{k+1}, on Δ′_{k}. Thus (8) holds at k + 1.

Suppose g ∈ G with T_{k+1}^{g} ≤ S. Then as T_{k} ≤ T_{k+1} and T_{k} is weakly closed in S, g ∈ N_{G}(T_{k}). Hence as P_{k} is the set of orbits of T_{k} on Ω, g ∈ N_{G}(P_{k}). The blocks of P′_{k} are of length less than 2^{k} by (5), while the blocks of P̄_{k} are of length 2^{k} by (4). Thus

N_{G}(P_{k}) ≤ N_{G}(P̄_{k}), so gP̄_{k} ∈ Sym(P̄_{k}) with (T P̄_{k} ; k+1)^{gP̄_{k}} ≤ SP̄_{k}. Hence by 5.2.3, gP̄_{k} acts

on T P̄_{k} ; k+1, so g acts on T_{k+1}, completing the proof of (7).

Finally assume (n, G) is not (2, L), (3, L), or (4, L). Then by 5.2.5, S = N_{G}(S). But as S ∈ Syl_{2}(G), N_{G}(S) is transitive on the S-invariant partitions of any given type, so (9) follows.

(5.7) Assume B ∈ B_{∗}(G, S) is transitive on Ω, and (n, G) ≠ (4, L). Then for some 0 ≤ k ≤ a(n) − 2:

(1) B = Z_{k}B_{Y}, where Z_{k} is the preimage in S of O_{2}(N_{Sym(P_{k})}(P_{k+2}/P_{k})), and B_{Y} ∈ Syl_{2}(Y) for some complement Y^{P_{k}} ≅ Syl(P_{k+2}) to Sym(P_{k})_{P_{k+2}/P_{k}} in N_{Sym(P_{k})}(P_{k+2}/P_{k}).

(2) N_{G}(B)/B ≅ S_{3}.

(3) N_{G}(P) ≤ N_{G}(P_{i}) for each i ≠ k + 1 with 1 ≤ i ≤ a(n).

Proof. Let Z_{B} = Ω_{1}(Z(B)) and Z the set of orbits of Z_{B} on Ω. As B is transitive on Ω:

(a) n = 2^{e}, where e = a(n), and Z is a N_{G}(B)-invariant regular (2^{a}, n/2^{a})-partition of Ω for some 1 ≤ a ≤ e.

For Δ ∈ Z, let Z(Δ) be the projection of Z_{B} on M_{Ω−Δ}, and set Z_{0} = ∏_{Δ∈Z} Z(Δ) and Z_{+} = Z_{0} ∩ G. As B = O_{2}(N_{G}(B)), Z_{+} ≤ B. Moreover Z(Δ) is elementary abelian and regular on Δ, so Z(Δ) is selfcentralizing in Sym(Δ). Then as N_{B}(Δ) centralizes Z(Δ), it follows that N_{B}(Δ)^{Δ} = Z(Δ).

Let K_{Δ} = G_{Ω−Δ}. Then N_{K_{Δ}}(Z(Δ)) = Z(Δ)X(Δ), with X(Δ) ≅ GL(Z(Δ)) ≅ GL_{a}(2). Let X = ∏_{Δ∈Z} X(Δ). Then N_{G}(B) acts on X_{0} = XZ_{+} and on X̄_{0} = X_{0}/Z_{+}, with X̄_{0} the direct product of the groups \overline{X(Δ)} ≅ X(Δ) for Δ ∈ Z. Then as N_{B}(Δ)^{Δ} = Z(Δ) and B is transitive on Ω, it follows that \overline{N_{X_{0}}(B)} = CX̄_{0}(B̄) is a full diagonal subgroup of X̄_{0} isomorphic to X(Δ). As N_{G}(B)/B has a strongly embedded subgroup, it follows that 2^{a} = |Δ| = 2 or 4, so that

(b) a = 1 or 2.

Let Z = Ω_{1}(Z(S)). As (n, G) ≠ (4, L), Z = ⟨z⟩, where z is the involution such that P is the set of orbits of ⟨z⟩ on Ω. (cf. 5.4.4) Now Z ≤ N_{S}(B) ∩ C_{S}(Z_{+}) = Z_{+} ≤ B, so Z ≤ Z_{B}. Therefore if a = 1 then Z = Z_{B} and Z = P, as P is the set of orbits of Z. We record this as:

(c) If a = 1 then Z_{B} = Z and Z = P.

Suppose for the moment that a = 2 and set K = G_{Z}. Then Z_{0} = Z_{+} = O_{2}(N_{G}(P_{2})), and Z_{B} ≅ Z(Δ) ≅ E_{4} is a full diagonal subgroup of Z_{0}. Further K/Z_{0} is of index at most 2 in the direct product of n/4 copies of S_{3}. As we saw above, N_{K}(B) = N_{K}(Z_{B}) with N_{K}(Z_{B})/Z_{0} ≅ S_{3}.

Next N_{M}(Z_{B}) = N_{K}(Z_{B})Y, where Y is a complement in N_{M}(Z_{B}) to M_{Z}, and Y acts faithfully on Z as Sym(Z). Further C_{M}(Z_{B}) = Z_{0}Y. Then as Z_{0} ≤ B ≤ C_{G}(Z_{B}), B = Z_{0}B_{Y}, where B_{Y} = B ∩Y. Then N_{G}(B) = N_{K}(Z_{B})N_{Y} (B_{Y}) with [N_{Y} (B_{Y}), N_{K}(Z_{B})] ≤ C_{K}(Z_{B}) = Z_{0} ≤ B. Therefore as N_{G}(B)/B has a strongly embedded subgroup, it follows that B_{Y} is Sylow in N_{Y} (B_{Y}), and hence also Sylow in Y. Then by 5.2.5, B_{Y} = N_{Y} (B_{Y}), so N_{G}(B) = N_{K}(Z_{B})B_{Y}.

Now N_{S}(B) acts on P_{2} with Z_{0} ≤ O_{2}(G_{P_{2}}), so Z_{0} = O_{2}(G_{P_{2}}) and hence P_{2} = Z as Z and P_{2} are the orbits of Z_{0} on Ω. Then as P_{i} ≤ P_{2} for i ≥ 2, K acts on P_{i}. Therefore N_{G}(B) = N_{K}(Z_{B})B_{Y} acts on P_{i}, Thus the lemma holds in this case with k = 0, so we may assume a = 1.

Now |P | = 2^{e−1}, so by induction on n, there exists an integer j with 0 ≤ j < e − 2 such that B^{P}, Sym(P), j satisfy the conclusions of then lemma in the roles of B, G, k. Set k = j + 1. Then the lemma holds for this choice of k.

(5.8) Assume ∅ ≠ Γ_{1} ⊂ Ω and set Γ_{2} = Ω − Γ_{1} and H = N_{G}(Γ_{1}). Assume T = S ∩ H ∈ Syl_{2}(H) and B ∈ B_{∗}(H, T). Set J = N_{H}(B), K_{i} = G_{Γ_{3−i}}, H_{i} = N_{K_{i}}(B), and K = K_{1}K_{2}. Let B_{i} be the projection of B ∩ K on K_{i}, and if G = L and n_{i} = |Γ_{i}| > 1 for i = 1 and 2, let h = h_{1}h_{2} ∈ H with h_{i} ∈ M_{Γ_{3−i}} an odd permutation of Γ_{i} for i = 1 and 2. Pick h ∈ B if B ≰ K, and pick h ∈ J if J ≰ K. Then

(1) B ∩ K = B_{1} × B_{2} with B_{i} ∈ B(K_{i}, S ∩ K_{i}).

(2) Either B = B ∩ K, or G = L and B = (B ∩ K)⟨h⟩.

(3) Either J = H_{1} × H_{2} or J ≰ K and J = H_{1}H_{2}⟨h⟩.

(4) Set J̄ = J/B. Then, up to a permutation of {1, 2}, one of the following holds:

(i) B_{1} ∈ B_{∗}(K_{1}, S∩K_{1}), H̄_{1} = J̄ has a strongly embedded subgroup, and B_{2} = S∩K_{2} = H_{2}.

(ii) h ∈ J − B, H̄_{1}⟨h̄⟩ = J̄ has a strongly embedded subgroup, and B_{2} = S ∩ K_{2} = H_{2}. (iii) h ∈ J − B, 3 ≤ n_{2} ≤ 5, B_{2} = S ∩ K_{2}, H̄_{2} ≅ Z_{3} is inverted by the involution h̄,

|H̄_{1}| is even, and H̄_{1}⟨h̄⟩ has a strongly embedded subgroup.

(iv) h ∈ J − B, and for i = 1 and 2, 3 ≤ n_{i} ≤ 5, B_{i} = S ∩ K_{i}, and H̄_{i} ≅ Z_{3}. Further H̄ is O(H̄) ≅ E_{9} extended by an involution h̄ inverting O(H̄). Proof. As B ∈ B(G, S), (1) follows from 8.2 in [A3]. If G = M or n_{i} = 1 for i = 1 or 2, then H = K and (2) and (3) are trivial. Otherwise, |H : K| = 2 with H = K⟨h⟩, and (2) and (3) follow from the choice of h.

Next O_{2}(H̄_{i}) = 1 by (1). Suppose H̄_{1} is of even order. Then as J̄ has a strongly embedded subgroup J̄_{0}, it follows that H̄_{2} ≤ CJ̄(\overline{H_{1} ∩ S}) ≤ J̄_{0}. Then by symmetry, H̄_{2} is of odd order, so B_{2} = S ∩ K_{2}. As H̄_{2} is of odd order and J̄_{0} is strongly embedded in J̄, J̄/H̄_{2} ≅ H̄_{1} or H̄_{1}⟨h̄⟩ has a strongly embedded subgroup.

Suppose B_{2} = H_{2}. Then either J̄ = H̄_{1}, or h ∈ J − B and J̄ = H̄_{1}⟨h̄⟩. Thus (i) or (ii) hold in this case, so we may assume that B_{2} ≠ H_{2}. Then by 5.2.5, G = L, h ∉ B, 3 ≤ n_{2} ≤ 5, and |H_{2} : B_{2}| = 3. But now (iii) holds in this case.

Hence we may assume |H̄_{i}| is odd for i = 1 and 2. Therefore B_{i} = S ∩ K_{i} for i = 1 and 2. If B_{i} = H_{i} for i = 1 and 2, then B = T = J, contradicting J̄_{0} strongly embedded in J̄. Hence by 5.2.5, we may assume 3 ≤ n_{1} ≤ 5, H̄_{1} ≅ Z_{3}, and h ∈ J − H. Now if n_{2} ∉ {3, 4, 5}, then by 5.2.5, H_{2} = B_{2} = S ∩ K_{2}, so that (ii) holds. Therefore we may assume 3 ≤ n_{2} ≤ 5, so that H̄_{2} ≅ Z_{3}. Then (iv) holds, completing the proof of the lemma.

(5.9) Assume B ∈ B_{∗}(G, S) is not transitive on Ω but N_{G}(B) is transitive on Ω. Assume (n, G, B) ≠ (5, L, 1). Then

(1) B has 3 orbits Γ_{i}, 1 ≤ i ≤ 3, on Ω, each of length 2^{k} for some nonnegative integer k. Thus n = 3 · 2^{k} = 2^{k+1} + 2^{k}.

(2) B = T_{k} (with T_{0} = 1) and N_{G}(B)/B ≅ S_{3}.

(3) We can choose notation so that Ω_{k+1} = Γ_{1} ∪ Γ_{2} and Ω_{k} = Γ_{3}.

(4) N_{G}(B) ≤ N_{G}(P_{i}) for 1 ≤ i ≤ k. Proof. Let Q = {Γ_{i} : 1 ≤ i ≤ r} be the set of orbits of B on Ω. As J = N_{G}(B) is transitive on Ω, but B is not, it follows that r > 1 and |Γ_{i}| = 2^{k} for some nonnegative integer k and each 1 ≤ i ≤ r.

If B = 1 then J = G, so G has a strongly embedded subgroup. Hence either n = 3 and G = M ≅ S_{3}, or n = 5 and G = L ≅ A_{5}. In the former case the lemma holds with k = 0, while the latter case is excluded by hypothesis. Thus we may assume B ≠ 1, so in particular k > 0. As J is transitive on Ω, Q is a regular (2^{k}, r)-partition of Ω and J ≤ N_{G}(Q). By construction, B ≤ G_{Q}. Set J̄ = J/B.

Now J_{Q} ⊴ J, so as J̄ has a strongly embedded subgroup J̄_{0}, either |J_{Q} : B| is odd, or, setting J_{Q,0} = J_{Q} ∩ J_{0}, we have J̄_{Q,0} strongly embedded in J̄_{Q} and J = J_{Q}J_{0}. Suppose the latter case holds. Let Γ′_{1} = Γ_{1} and Γ′_{2} = Ω−Γ_{1}. Applying 5.8 to Γ′_{1} in the role of “Γ_{1}”, we conclude from 5.8.4 that B′_{i} = B_{Γ′_{3−i}} is Sylow in K′_{i} = N_{G}(Γ_{1})_{Γ′_{3−i}} for i = 1 or 2, and that B = B′_{1}B′_{2} or B′_{1}B′_{2}⟨h⟩ for some h ∈ J_{Q} −K′_{1}K′_{2}. For 1 ≤ j ≤ r, set K_{j} = G_{Ω−Γ_{j}}, so that K_{1} = K′_{1}. As J is transitive on Q, it follows that B_{j} = B ∩ K_{j} ∈ Syl_{2}(K_{j}) for each 1 ≤ j ≤ r. Then if k ≠ 2 or B ≠ C_{B}(K_{1})B_{1}, it follows from 5.2.5 that N_{K_{j}}(B) = B_{j} for each 1 ≤ j ≤ r, so that J_{Q} is a 2-group, contradicting J̄_{Q,0} strongly embedded in J̄_{Q}. Therefore k = 2 and B = C_{B}(K_{1})B_{1}. Hence B = O_{2}(G_{Q}) and J = N_{G}(Q). But then J̄ does not have a strongly embedded subgroup, a contradiction.

Therefore |J_{Q} : B| is odd, so that B ∈ Syl_{2}(J_{Q}), and then B ∈ Syl_{2}(G_{Q}). Then, using 5.2.5, J_{Q} = B. Now N_{G}(Q) = G_{Q}J with J̄ ≅ J^{Q} = Sym(Q), so as J̄ has a strongly embedded subgroup, it follows that r = |Q| = 3, completing the proof of (1).

Next for 1 ≤ i ≤ k, it follows from 5.6 that there exists a regular B-invariant (2^{i}, n/2^{i})- partition Q_{i} of Ω such that Q ≤ Q_{i} ≤ Q_{j} for j ≤ i, and such that T′_{i} ≤ B, where T′_{i} is a Sylow 2-subgroup of G_{Q_{i}} with T′_{k} = B. Let B ≤ S′ ∈ Syl_{2}(N_{G}(Q_{k})). Then Q_{i} = P_{i}(S′, Ω) by 5.6.9, so by 5.6.8, T′_{i} is weakly closed in S′. Then as T′_{i} ≤ B ≤ S, it follows that T_{i} = T′_{i}. Then P_{i} = Q_{i} by 5.6.9 and T_{k} = T′_{k} = B, completing the proof of (2). As T_{i} is weakly closed in S, N_{G}(B) ≤ N_{G}(T_{i}), and then N_{G}(B) ≤ N_{G}(P_{i}) by 5.6.8, establishing (4).

Finally we may choose notation so that N_{S}(B) = B⟨s⟩, where s interchanges Γ_{1} and Γ_{2} and fixes Γ_{3}. Thus Ω′_{k+1} = Γ_{1} ∪ Γ_{2} and Ω′_{k} = Γ_{3} are orbits of N_{S}(B) on Ω of length 2^{k+1} and 2^{k}, respectively. But by (1) and 5.4.1, S has orbits Ω_{i} of length 2^{i} for i ∈ {k, k + 1}, so (3) follows.

(5.10) Assume B ∈ B_{∗}(G, S) and F ix(B) ≠ ∅. Then one of the following holds: (1) n is odd, F ix(B) = {α} = Ω_{0} is of order 1, S ≤ G_{α} ≥ N_{G}(B), and B ∈ B_{∗}(G_{α}, S).

(2) n ≡ 2 mod 4, |F ix(B)| = 2, G = L, and F ix(B) = Ω_{1}.

(3) n is odd, |F ix(B)| = 3, and one of the following holds:

(i) n = 7, G = L, B ≅ E_{4}, Mov(B) = Ω_{2}, and N_{G}(B)/B is E_{9} extended by an involution inverting O(N_{G}(B)/B).

(ii) G = L, n ≡ 3 mod 4, and F ix(B) = Ω_{1} ∪ Ω_{0}.

(iii) B ∈ Syl_{2}(G_{F ix(B)}), B = N_{G}(B)_{F ix(B)}, and N_{G}(B)/B ≅ S_{3}. Moreover either N_{G}(B) ≤ N_{G}(P_{i}^{∘}) for some 2 ≤ i ≤ a(n), or n = 2^{e} + 1 for some e ≥ 1.

(4) n = 5 = |F ix(B)|, B = 1, and G = L.

Proof. Let F = F ix(B) and m = |F |. Observe n ≡ m mod 2, so n is odd iff m is odd.

Suppose first that m = 1, so that F = {α} for some α ∈ Ω. Then N_{G}(B) ≤ G_{α} and as B ≤ S and S is nilpotent, S ≤ G_{α}. Then as B ∈ B_{∗}(G, S), also B ∈ B_{∗}(G_{α}, S). By 5.4.1, F = Ω_{0}. That is (1) holds in this case.

Assume next that m = 2, If G = M then the transposition t with Mov(t) = F is in O_{2}(N_{G}(B)) ≤ B, contradicting F ⊈ F ix(t). Hence G = L. If Δ is an orbit of B of length 2, then the involution s with Mov(s) = F ∪ Δ acting on F is in O_{2}(N_{G}(B)) ≤ B, again a contradiction. Thus all nontrivial orbits of B are of length at least 4, so n ≡ m = 2 mod 4, and as B ≤ S, F = Ω_{1} is the unique orbit of S of length 2. Thus (2) holds in this case.

Therefore we may assume m > 2. Let K = G_{Ω−F}. As K ⊴ N_{G}(B) and N_{G}(B)/B has a strongly embedded subgroup, either K is of odd order or K has a strongly embedded subgroup. We conclude that either:

(a) m = 3, K ≅ A_{3} or S_{3}, and if K ≅ A_{3} then G = L, or

(b) m = 5, K ≅ A_{5}, and G = L.

Suppose K is of even order. Then K has a strongly embedded subgroup, so applying 5.8 with F = Γ_{1}, we conclude that B = S_{F} ∈ Syl_{2}(G_{F}). In particular if (b) holds and F ≠ Ω, then there is an involution s ∈ N_{S}(B) which is an odd permutation on F and Ω − F, so that N_{S}(B)/B ≅ S_{5} does not have a strongly embedded subgroup. Thus in case (b), Ω = F, so that (4) holds in this case.

Therefore we may assume case (a) holds, so that m = 3. Suppose that N_{S}(B) ≤ G_{F}. Then K is of odd order, so G = L. Arguing as in the case m = 2, B has no orbits of length 2, so n ≡ 3 mod 4, and then as B ≤ S, F = Ω_{1} ∪ Ω_{0}. That is (3ii) holds in this case.

Therefore we may assume N_{G}(B)/N_{G}(B)_{F} ≅ N_{G}(B)^{F} = Sym(F) ≅ S_{3}. Set X = N_{G}(B)_{F}. We apply 5.8 with Γ_{1} = F to conclude that either B ∈ Syl_{2}(G_{F}), or conclusion (iii) of 5.8.4 holds. Assume the latter, and set H = X⟨h⟩ and H̄ = H/B. If B has an orbit {β, γ} of length 2, then we may choose h to be an involution with cycle (β, γ), moving four points of Ω. But now as |X̄| is even, m_{2}(H̄) > 1, contradicting |H̄ : X̄| = 2 and H̄ possessing a strongly embedded subgroup. Thus all nontrivial orbits of B are of length at least 4, so n ≡ 3 mod 4, and (3ii) holds. Thus B ∈ Syl_{2}(G_{F}) and by 5.8.4, either B = X or 3 ≤ n − 3 ≤ 5. Assume the latter. By paragraph one, n is odd, so n − 3 = 4 and hence n = 7. As B ≠ X, it follows that B ≅ E_{4}, and then that (3i) holds.

Therefore we may assume that B = X, so that N_{G}(B)/B ≅ S_{3} by the previous paragraph. Suppose n ≠ 2^{e} + 1 for some e ≥ 1. Then

(*)  n − 3 ≠ 2(2^{e−1} − 1) = 2^{e} + 2^{e−1} + ⋯ + 2.

Write n − 3 = ∑_{i=1}^{∞} b_{i}2^{i} with b_{i} ∈ {0, 1} in binary expansion. Then by (*) there exists an i ≥ 1 with b_{i} = 0 and b_{i+1} = 1. Pick the least such i. If b_{1} ≠ 0 then

n ≡ ∑_{j=1}^{i−1} b_{j}2^{j} + 3 = 2(2^{i−1} − 1) + 3 = 2^{i} + 1 mod 2^{i+1},

so from 5.3 and 5.4, and as B ≤ S is Sylow in G_{F}, if follows that F ix(N_{S}(B)) = Ω_{0}, Ω_{i} is the union of the nontrivial orbits of N_{S}(B) of length less than i, and for j > i, each orbit of S of length 2^{j} is an orbit of B. In particular Δ̄_{i+1} is the union of the orbits of B (and of S) of length at least 2^{i+1}, so N_{G}(B) ≤ N_{G}(P_{i+1}^{∘}). That is (3iii) holds in this case.

On the other hand if b_{1} = 0 then n ≡ 3 mod 2^{i+1}, and again Δ̄_{i+1} is the union of the orbits of B and S of length at least 2^{i+1}, and once again (3iii) holds.

(5.11) Assume n is even, B ∈ B_{∗}(G, S), and (n, G) ≠ (4, L). Then N_{G}(B) ≤ N_{G}(P_{i}) for i = 1 or 2.

Proof. Assume otherwise. By 5.7.3, B is not transitive on Ω. Then by 5.9, N_{G}(B) is not transitive on Ω.

Suppose F ix(B) ≠ ∅. Then by 5.10, n ≡ 2 mod 4 and F ix(B) = Ω_{1}. Applying 5.8 to Γ_{2} = Ω_{1}, and setting Γ = Ω − Ω_{1}, we have B^{Γ} ∈ B_{∗}(Sym(Γ), S^{Γ}), and by induction on n, the lemma is satisfied by B^{Γ}, Sym(Γ) in the roles of “B, G”. Thus for i = 1 or 2, N_{Sym(Γ)}(B^{Γ}) ≤ N_{Sym(Γ)}(P_{i}(Sym(Γ)), S^{Γ}), so as Γ_{2} = Ω_{1}, it follows that N_{G}(B) ≤ N_{G}(P_{i}).

Thus we may assume F ix(B) = ∅, so all orbits of B on Ω are nontrivial. Let 2^{a} be the minimal length of an orbit of B, and let Γ_{1} be an orbit of N_{G}(B) on Ω which is the union of r B-orbits of length 2^{a}. As N_{G}(B) is not transitive on Ω, ∅ ≠ Γ_{1} ⊂ Ω. Thus we can apply 5.8 to Γ_{1}, and adopt the notation of that lemma. Note that n_{i} = |Γ_{i}| ≡ 0 mod 2^{a}.

Suppose N_{G}(B)^{Γ_{1}}/B^{Γ_{1}} has a strongly embedded subgroup. Then by 5.8.4, B_{2} = S_{Γ_{1}} ∈ Syl_{2}(G_{Γ_{1}}), and either B_{2} = N_{G}(B)_{Γ_{1}} or n_{2} = 4 and G = L. Suppose further that r ≠ 1. Then by 5.9, N_{G}(B)^{Γ_{1}} ≤ N_{Sym(Γ_{1})}(P_{i}(S^{Γ_{1}}, Γ_{1})) for i ≤ a. Thus if B_{2} = N_{G}(B)_{Γ_{1}} then N_{G}(B) ≤ N_{G}(P_{1}) and the lemma holds, so assume otherwise. Then we saw that n_{2} = 4, so if a > 1 then N_{G}(B) ≤ N_{G}(P_{2}) and the lemma holds. Finally if a = 1 then n = 10 and N_{G}(B)/B ≅ S_{3} × S_{3} does not have a strongly embedded subgroup, a contradiction.

Therefore r = 1. Hence by 5.7, N_{Sym(Γ_{1})}(B^{Γ_{1}}) ≤ N_{Sym(Γ_{1})}(P_{i}(Sym(Γ_{1})), Γ_{1})) for each i ≠ k + 1, and some nonnegative integer k ≤ a − 2. Hence if B_{2} = N_{G}(B)_{Γ_{1}} then as n_{2} ≥ 2^{a}, it follows that N_{G}(B) ≤ N_{G}(P_{i}) for i = 1 or 2. So we may assume B_{2} ≠ N_{G}(B)_{Γ_{1}} and hence n_{2} = 4 and G = L. Then as n_{2} ≥ 2^{a}, it follows that a = 2, n = 8, and N_{G}(B) is transitive on Ω, contrary to an earlier reduction.

We have shown that N_{G}(B)^{Γ_{1}}/B^{Γ_{1}} does not have a strongly embedded subgroup. Then by 5.8, B_{1} = B_{Γ_{2}} = S_{Γ_{2}} ∈ Syl_{2}(G_{Γ_{2}}). In particular by 5.3 and 5.4.1, B has a unique orbit of length 2^{a} on Γ_{1}, so r = 1. Therefore each orbit Γ of B of length 2^{a} is N_{G}(B)-invariant, and B_{Ω−Γ} = S_{Ω−Γ}. It follows that B has a unique orbit Γ of length 2^{a}.

As B is not transitive on Ω, Γ_{1} = Ω − Γ is nonempty, and by uniqueness of Γ, all orbits of B on Γ_{1} have length at least 2^{a+1}. Therefore n_{1} = |Γ_{1}| ≡ 0 mod 2^{a+1}. By 5.8, N_{Sym(Γ_{1})}(B^{Γ_{1}})/B^{Γ_{1}} has a strongly embedded subgroup, so by induction on n, N_{Sym(Γ_{1})}(B^{Γ_{1}}) ≤ N_{Sym(Γ_{1})}(P_{i}(Γ_{1}, S^{Γ_{1}})) for i = 1 or 2. Therefore if a ≥ i then N_{G}(B) ≤ N_{G}(P_{i}), and the lemma holds. It follows that a = 1 and i = 2. Therefore n ≡ 2 mod 4, and as all orbits of B on Γ_{1} are of length at least 4, Γ = Ω_{1}. But then N_{G}(B) ≤ N_{G}(P_{2}), so the proof is at last complete.

Theorem 5.12. Assume B ∈ B_{∗}(G, S). Then one of the following holds:

(1) n = 4, G = M, B = O_{2}(M), and N_{G}(B)/B ≅ S_{3}.

(2) n = 5, B = 1, and G = L ≅ A_{5}.

(3) ⟨N_{G}(B), S⟩ is a proper subgroup of G.

(4) n = 2^{e} + 1 for some e ≥ 2, |F ix(B)| = 3, B = N_{G}(B)_{F ix(B)} ∈ Syl_{2}(G_{F ix(B)}), and N_{G}(B)/B ≅ S_{3}.

Proof. Let H = ⟨N_{G}(B), S⟩. Assume first that n is even. If n = 2 then B_{∗}(G, S) = ∅. If n = 4 then B_{∗}(G, S) = ∅ when G = L, while B_{∗}(G, S) = {O_{2}(G)} if G = M. Hence we may assume n ≥ 6. Then by 5.11, N_{G}(B) ≤ N_{G}(P_{i}) for i = 1 or 2. As n ≥ 6, N_{G}(P_{i}) is a proper subgroup of G, and by 5.6.2, S ≤ N_{G}(P_{i}), so H ≤ N_{G}(P_{i}). Therefore (3) holds in this case.

Therefore we may assume that n is odd. Therefore F = F ix(B) ≠ ∅. If F = {α} is of order 1, then by 5.10, S ≤ G_{α} ≥ N_{G}(B), so that H ≤ G_{α} < G, and hence (3) holds in this case. Similarly if the conclusions of 5.10.4 hold, conclusion (2) of the theorem is satisfied. Hence by 5.10, we may assume that |F | = 3, and that one of the three cases of 5.10.3 are satisfied. Indeed conclusion (3) or (4) of the theorem holds in case (iii) of 5.10.3, so we may assume case (i) or (ii) holds. In case (i), H ≤ N_{G}(Ω_{2}) < G. Finally in case (ii), H ≤ N_{G}(Ω_{1} ∪ Ω_{0}), so once again (3) holds. This completes the proof of the theorem.

Section 6. Generation of fusion systems

In this section p is a prime and F is a saturated fusion system on a finite p-group S. Recall from Section 2 that if U ∈ F^{f} such that N_{F}(U) is constrained, then G(U) denotes a model of N_{F}(U). Thus G = G(U) is a finite group with Sylow p-subgroup S(G) = N_{S}(U), F^{∗}(G) = O_{p}(G), and N_{F}(U) = F_{S(G)}(G).

Recall Notation 2.3, and let Ω be an F-invariant collection of subgroups of S.

Hypothesis 6.1. B is a subsystem of F on S containing Aut_{F}(S). Let Λ = Λ(B) be the set of pairs (P, ϕ) such that P ≤ S, ϕ ∈ hom_{F}(P, S), and ϕ ∉ hom_{B}(P, S). Let Λ^{∗} = Λ^{∗}(B) consist of those (P, ϕ) ∈ Λ such that |P | is maximal subject to this constraint. Let U = U(B) be the set of U ∈ F^{frc} such that Aut_{F}(U) ≠ Aut_{B}(U), and |U| is maximal subject to this constraint.

(6.2) Assume Hypothesis 6.1 with B ≠ F and let (P, ϕ) ∈ Λ^{∗} and α ∈ 𝔄(P). Then (1) P ≠ S.

(2) α ∈ hom_{B}(N_{S}(P), S) and (P α, βϕ) ∈ Λ^{∗}, where β : P α → P is the inverse of α : P → P α.

(3) P α ∈ U, and if F is Ω-Alperin then P α ∈ F_{Ω}^{f}.

(4) For each U ∈ U, |U| = |P |, and for each ϕ ∈ Aut_{F}(U) − Aut_{B}(U), (U, ϕ) ∈ Λ^{∗}. If F is Ω-Alperin then U ∈ F_{Ω}^{f}.

(5) Out_{B}(P α) is a strongly p-embedded subgroup of Out_{F}(P α).

(6) For each U ∈ U and each subgroup R of N_{S}(U) with U < R, Aut_{F}(R) = Aut_{B}(R).

Proof. If P = S then ϕ ∈ Aut_{F}(P) and as Aut_{F}(S) ⊆ B by hypothesis, we have Aut_{F}(P) = Aut_{B}(P), contrary to the choice of (P, ϕ) ∈ Λ. Thus (1) holds.

By (1), P < N_{S}(P), so by maximality of |P |, α : N_{S}(P) → N_{S}(P α) is a B-map. Thus β ∈ hom_{B}(P α, P). But ϕ = β^{−1} · βϕ, so if βϕ is a B-map then so is ϕ, a contradiction. Hence (2) holds.

By Alperin’s Fusion Theorem (cf. A.10 in [BLO]), there exist U_{i} ∈ F^{frc}, β_{i} ∈ Aut_{F}(U_{i}), and P_{i} ≤ U_{i}, 1 ≤ i ≤ n, such that ϕ = β_{1} ⋯ β_{n}, P = P_{1}, and P_{i+1} = P_{i}β. Thus as ϕ is not a B-map, there exists some i such that β_{i} : P_{i} → P_{i+1} is not a B-map. If P_{i} ≠ U_{i}, then β_{i} ∈ Aut_{B}(U_{i}) by maximality of |P |, contrary to the choice of i. Hence P_{i} = U_{i} ∈ F^{frc} and Aut_{F}(P_{i}) ≠ Aut_{B}(P_{i}). As P_{i} ∈ P^{F} = (P α)^{F}, there exist γ ∈ 𝔄(P_{i}) with P_{i}γ = P_{i}α. Then Aut_{F}(P_{i})γ^{∗} = Aut_{F}(P α), and by (2) γ : N_{S}(P_{i}) → N_{S}(P α) is a B-map, so Aut_{B}(P_{i})γ^{∗} = Aut_{B}(P α). Moreover if F is Ω-Alperin, then we can choose U_{i} ∈ F_{Ω}^{f}, so P_{i} and P α are also in F_{Ω}^{f}. This completes the proof of (3).

By (3) and definition of U, |P | = |U| for each U ∈ U, so (4) follows. Part (6) is immediate from (4) and the maximality of |P |.

Finally we prove (5). Let U = P α, G = G(U), and Ḡ = G/U. Then Aut_{F}(U) = Aut_{G}(U) ≅ G/Z(U), and Out_{F}(U) ≅ Ḡ. Let H be the preimage in G of Aut_{B}(U). By (3), U ∈ U, so H̄ ≠ Ḡ. As U is fully normalized, T = N_{S}(U) ∈ Syl_{p}(G). By 46.4 in [FGT], it suffices to show that for each 1 ≠ X ≤ T̄, NḠ(X) ≤ H̄. Now X = R̄ for some U < R ≤ T, and NḠ(X) = \overline{N_{G}(R)}. By (6), Aut_{F}(R) = Aut_{B}(R). Thus for g ∈ N_{G}(R), c_{g|R} ∈ Aut_{B}(R), so ϕ = c_{g|U} ∈ Aut_{B}(U), and hence the image ḡ of ϕ in Out_{F}(U) = Ḡ is in Out_{B}(U) = H̄, completing the proof of (5).

Hypothesis 6.3. F is a local CK-system and B is a subsystem of F on S containing Aut_{F}(S). Further F is an Ω-system.

For U ∈ U = U(B), let W(U) be the set of W ∈ F^{f} such that N_{S}(U) ≤ N_{S}(W), Aut_{N_{F} (W)}(U) ≠ Aut_{N_{B}(W)}(U), and |N_{S}(W)| is maximal subject to this constraint. By 6.2.4, U ∈ F_{Ω}^{f}, so as N_{S}(U) ≤ N_{S}(W), also W ∈ F_{Ω}^{f}. Then as F is an Ω-system, N_{F}(W) is constrained.

Let Y(W, U) be the set of subgroups Y of G = G(W) such that S ∩ Y ∈ Syl_{p}(Y), U ≤ N_{S∩Y} (U) ∈ Syl_{p}(N_{Y} (U)), and Aut_{Y} (U) ≰ Aut_{B}(U). Let X (W, U) consist of those X ∈ Y(W, U) such that N_{S}(W) ≤ X. Let Z = Ω_{1}(Z(S)) and for Y ∈ Y(W, U) let V (Y) = ⟨Z^{Y} ⟩.

Assume B ≠ F and pick P ∈ U and W ∈ W(P). Set Y = Y(W, P), X = X (W, P), T = N_{S}(W), and G = G(W).

(6.4) Assume Hypothesis 6.3. Then (1) W ∈ F_{Ω}^{f}, F^{∗}(G) = O_{p}(G), T ∈ Syl_{p}(G), and N_{F}(W) = F_{T} (G).

(2) For each Y ∈ Y, Z ≤ O_{p}(Y) ≤ O_{2}(N_{Y} (P)) = P, Out_{B}(P) ∩ Out_{Y} (P) is strongly p-embedded in Out_{Y} (P) = N_{Y} (P)/P, and P ∈ B_{∗}(Y, S ∩ Y).

(3) P ∈ N_{F}(W)^{frc}, N_{S}(P) = N_{T} (P) is Sylow in N_{G}(P), N_{N_{F} (W)}(P) = F_{N_{T} (P)}(N_{G}(P)), and P = O_{p}(N_{G}(P)).

(4) For each X ∈ X, Z ≤ O_{p}(X) ≤ P.

(5) W Z ≤ O_{p}(G) ≤ P. Proof. Let N = N_{F}(W), R = N_{S}(P), and H = N_{G}(P). We observed in 6.3 that W ∈ F_{Ω}^{f} and N is constrained, so the model G = G(W) exists, and by definition F^{∗}(G) = O_{p}(G). Thus (1) holds.

Let Y ∈ Y and suppose O_{p}(Y) ≰ P. As P ≤ T_{Y} = T ∩ Y ∈ Syl_{p}(Y), O_{p}(Y) ≤ T. As O_{p}(Y) ≰ P, N_{O_{p}(Y)}(P) ≰ P, so P < O_{p}(N_{Y} (P)) = I. Then by 6.2.6, Aut_{F}(I) = Aut_{B}(I), so Aut_{Y} (P) ≤ Aut_{B}(P), contrary to the choice of Y ∈ Y. Thus O_{p}(Y) ≤ P. In particular as G ∈ Y, O_{p}(G) ≤ P, and as N_{Y} (P) ∈ Y, P = O_{p}(N_{Y} (P)). As W ∈ F^{f}, Z ≤ Z(T), and then Z ≤ O_{p}(G) by (1), so that Z ≤ P as O_{p}(G) ≤ P.

As F^{∗}(G) = O_{p}(G) and O_{p}(G) ≤ P ≤ Y, F^{∗}(N_{Y} (P)) = O_{p}(N_{Y} (P)) = P. In particular C_{Y} (P) ≤ P, so Out_{Y} (P) = N_{Y} (P)/P C_{Y} (P) = N_{Y} (P)/P. As Y ∈ Y, Σ = Out_{B}(P)∩Out_{Y} (P) ≠ Out_{Y} (P), so by 6.2.5, Σ is strongly p-embedded in Out_{Y} (P). As Y ∈ Y, N_{T_{Y}} (P) ∈ Syl_{p}(N_{Y} (P)), so P ∈ B_{∗}(Y, S ∩ Y). This completes the proof of (2).

As P ∈ F^{f}, |N_{S}(P)| ≥ |N_{S}(P′)| for each P′ ∈ P^{F}, so P ∈ N^{f} and R = N_{T} (P). As P ∈ F^{fc}, C_{S}(P) ≤ P, so P ∈ N^{fc}. By (1), N = F_{T} (G), so as P ∈ N^{fc}, R ∈ Syl_{p}(H) and N_{N} (P) = F_{R}(H). As H ∈ Y, P = O_{p}(H) by (2), so P ∈ N^{frc}, completing the proof of (3). Similarly X ⊆ Y, so (2) implies (4). As G ∈ X, (4) implies (5).

(6.5) Assume Hypothesis 6.3, Y ∈ Y, and X ⊴ Y such that Aut_{X}(P) ≤ Aut_{B}(P). Set V = V (Y) and Y^{∗} = Y/C_{Y} (V). Assume C_{F}(Z) ≤ B. Then

(1) F^{∗}(Y) = O_{p}(Y), V is an elementary abelian p-group, and O_{p}(Y^{∗}) = 1.

(2) T ∩ X ≤ P and N_{Y} (T ∩ X) ∈ Y.

(3) F_{S∩Y} ((S ∩ Y)C_{Y} (V)) ≤ B, so R = C_{S∩Y} (V) ≤ P and N_{Y} (R) ∈ Y.

(4) Y = C_{Y} (V)N_{Y} (R), so Y^{∗} = N_{Y} (R)^{∗}. Proof. By 6.4.1, F^{∗}(G) = O_{p}(G), and by 6.4.5, O_{p}(G) ≤ P. As Y ∈ Y, P ≤ Y, so F^{∗}(G) ≤ Y ≤ G, and hence F^{∗}(Y) = O_{p}(Y). By 6.4.2, Z ≤ O_{p}(Y). As Z ≤ Z(S) and S ∩ Y ∈ Syl_{p}(Y), Z ≤ Z(O_{p}(Y)), so V = ⟨Z^{Y} ⟩ is elementary abelian, and by a standard argument, O_{p}(Y^{∗}) = 1. Hence (1) holds.

Set K = N_{Y} (P). As Y ∈ Y, S ∩ K ∈ Syl_{p}(K). Then as X ⊴ Y, S ∩ X ∩ K ∈ Syl_{p}(K ∩ X).

As Y ∈ Y there exists g ∈ K such that c_{g} ∉ Aut_{B}(P). As S ∩X ∩K ∈ Syl_{p}(K ∩X), it follows from a Frattini argument that g = xy for some x ∈ K ∩X and y ∈ N_{Y} (S∩X ∩K). Again by hypothesis, c_{x} ∈ Aut_{B}(P), so as c_{g} = c_{x}c_{y}, c_{y} ∉ Aut_{B}(P). Therefore by 6.2.6, S ∩ X ∩ K ≤ P, so T ∩ X ≤ P, and hence T ∩ X = S ∩ X ∩ K, completing the proof of (2).

As C_{F}(Z) ≤ B, the first statement in (3) follows. Then the remaining statements follow from (2) applied to C_{Y} (V) in the role of X, completing the proof of (3). Part (4) follows from a Frattini argument and (1).

(6.6) Assume Hypothesis 6.3, with C_{F}(Z) ≤ B and N_{F}(Baum(S)) ≤ B. Assume X ∈ X and set V = V (X), X^{∗} = X/C_{X}(V), and J(X) = ⟨J(T)^{X}⟩. Then

(1) For each 1 ≠ W_{1} ≤ S with W_{1} ⊴ X, W_{1} ∈ W(P) and T = N_{S}(W_{1}).

(2) If T ≠ S then no nontrivial characteristic subgroup of T is normal in X.

(3) J(T) ≰ C_{X}(V), so V is a failure of factorization module for X^{∗}.

(4) N_{J(X)}(C_{T} (V))T ∈ X.

Proof. Assume the hypothesis of (1), and let α ∈ 𝔄(W_{1}). As N_{S}(P) ≤ T and P ∈ F^{frc}, also P α ∈ F^{frc} with N_{S}(P α) ≤ N_{S}(W_{1}α). Thus β = α_{|N_{S}(P)} ∈ 𝔄(P), so by 6.2.2, P α ∈ U and W_{1}α ∈ W(P α). In particular, |N_{S}(W_{1}α)| = |T | by maximality of T, so T = N_{S}(W_{1}) and W_{1} ∈ F^{f}. This establishes (1). Then (1) implies (2).

Let R = C_{T} (V). By parts (3) and (4) of 6.5, N_{X}(R) ∈ Y with X = N_{X}(R)C_{X}(V). Then also N_{X}(R) ∈ X, and as X = N_{X}(R)C_{X}(V), we have V = V (N_{X}(R)). Hence replacing X by N_{X}(R), we may assume R ⊴ X. Suppose J(T) ≤ C_{X}(V). Then as R ⊴ X and R ∈ Syl_{p}(C_{X}(V)), we have J(T) ≤ R and then also J(T) ⊴ X. If T ≠ S this is contrary to (2). Hence T = S and J(S) ≤ C_{S}(V), so V ≤ Z_{J} = Ω_{1}(Z((J(S)))). Thus Baum(S) = C_{S}(Z_{J}) ≤ C_{S}(V) = R ⊴ X, so as Baum(S) is weakly closed in S with respect to F, it follows that Baum(S) ⊴ X. Then by hypothesis, F_{S}(X) ≤ B, contradicting X ∈ X. This establishes (3).

Next J(X) ⊴ X, so if Aut_{J(X)}(P) ≤ Aut_{B}(P) then by 6.5.2, N_{X}(S ∩ J(X)) ∈ Y. Then N_{X}(S∩J(X)) ∈ X with J(T) ≤ S∩J(X), contrary to (2). Therefore Aut_{J(X)}(P) ≰ Aut_{B}(P), so (4) holds.

Section 7. Generating 2-Fusion systems

In this section we assume:

Hypothesis 7.1. F is a saturated fusion system on a finite 2-group S. Assume Hypothesis 6.3 with p = 2. In addition assume that C_{F}(Z) ≤ B and N_{F}(Baum(S)) ≤ B. Write X_{∗} for the minimal members of X under inclusion, and pick X ∈ X_{∗}. Set V = V (X) and X^{∗} = X/C_{X}(V).

(7.2) (1) F^{∗}(X) = O_{2}(X), V is an elementary abelian 2-group and O_{2}(X^{∗}) = 1. (2) For each 1 ≠ W_{1} ⊴ X, W_{1} ∈ W(P) and T = N_{S}(W_{1}).

(3) If T ≠ S then no nontrivial characteristic subgroup of T is normal in X.

(4) J(T) ≰ C_{X}(V), so V is a failure of factorization module for X^{∗}.

(5) C_{X}(V) is 2-closed with C_{T} (V) = O_{2}(X) ≤ P and Aut_{C_{X}(V)}(P) ≤ Aut_{B}(P).

(6) X = J(X)T.

Proof. Part (1) follows from 6.5.1. Parts (2), (3), and (4) follow from parts (1)-(3) of 6.6. Part (5) follows from 6.5.3 and the minimality of X. Similarly (6) follows from 6.6.4 and the minimality of X.

Recall the definition of a minimal parabolic in Definition 1.1.

(7.3) (1) N_{X^{∗}}(P^{∗}) = N_{X}(P)^{∗}.

(2) If T ≤ M ≤ X with X = MC_{X}(V) then X = M.

(3) If X^{∗} is a minimal parabolic, then X is a minimal parabolic.

Proof. Pick M as in (2), set Y = C_{X}(V) and K = C_{M}(V). Let g ∈ M with g^{∗} ∈ N_{X^{∗}}(P^{∗}). Then g acts on P K, and by 7.2.5, P ∈ Syl_{2}(P K), so by a Frattini argument, g = mk with k ∈ K and m ∈ N_{M}(P). Thus g^{∗} = m^{∗}, so N_{M^{∗}}(P^{∗}) = N_{M}(P)^{∗}. Thus (1) holds by applying this fact with M = X.

Next as X ∈ X there is h ∈ N_{X}(P) with c_{h} ∉ Aut_{B}(P). As X = MY, we have M^{∗} = X^{∗}, so h^{∗} ∈ N_{X^{∗}}(P^{∗}) = N_{M^{∗}}(P^{∗}) = N_{M}(P)^{∗}. Thus h = my with m ∈ N_{M}(P) and y ∈ Y. Then y = m^{−1}h ∈ N_{X}(P), so c_{y} ∈ Aut_{B}(P) by 6.5.3. Then as c_{h} = c_{m}c_{y}, it follows that c_{m} ∉ Aut_{B}(P), so M ∈ X. Therefore M = X by minimality of X, establishing (2).

Assume X^{∗} is a minimal parabolic. Write M for the set of minimal parabolics of X over T. By McBride’s Theorem 1.2.2, X = ⟨M, N_{X}(T)⟩, so X^{∗} = ⟨Y^{∗}, N_{X}(T)^{∗} : Y ∈ M⟩. But as X^{∗} is a minimal parabolic, there is a unique maximal overgroup L^{∗} of T^{∗} in X^{∗}. Then as O_{2}(X^{∗}) = 1 by 7.2.1, N_{X}(T)^{∗} ≤ L^{∗}, and for Y ∈ M, either Y^{∗} ≤ L^{∗} or Y^{∗} = X^{∗}. It follows that there is some M ∈ M with X^{∗} = M^{∗}. Then X = M by (2), establishing (3).

(7.4) Assume X is not solvable. Then

(1) X^{∗} = E(X^{∗})T^{∗} and T^{∗} is transitive on the set L = {L_{1}^{∗}, …, L_{r}^{∗}} of components of X^{∗}.

(2) Each component L_{i}^{∗} is normal in J(X)^{∗}.

(3) L_{i}^{∗} is an image of SL_{n}(q), Sp_{n}(q), Spin_{n}(q), SU_{m}(q), m > 3, G_{2}(q), with q even, Â_{6}, or an alternating group.

(4) Let O_{1} be a P -invariant subset of L, and set O_{2} = L − O_{1} and let Y_{i} be the preimage in X of ⟨O_{i}⟩. Then for i = 1 or 2, Aut_{Y_{i}}(P) ≰ Aut_{B}(P), Y_{i}P ∈ Y, and Y_{3−i} ∩ T ≤ P.

(5) Let P_{i} be the preimage in T of the projection of P^{∗} ∩ E(X^{∗}) on L_{i}^{∗} and Q_{i} = N_{P} (L_{i}). Then P_{i} ≤ P, P_{i} = O_{2}(N_{L_{i}}(P_{i})), P_{i}^{∗} = O_{2}(N_{L_{i}^{∗}}(P_{i}^{∗})), Q_{i} = O_{2}(N_{L_{i}Q_{i}}(Q_{i})), and if L_{i}^{∗} is of Lie type of characteristic 2, then N_{L_{i}^{∗}}(P_{i}^{∗}) is a parabolic subgroup of L_{i}^{∗} with radical P_{i}^{∗}.

(6) There exists an orbit O_{1} of P on L such that, setting O_{2} = L − O_{1}, and writing Y_{i} for the preimage in X of ⟨O_{i}⟩, Aut_{Y_{1}}(P) ≰ Aut_{B}(P), Y_{1}P ∈ Y, and Y_{2} ∩ T ≤ P.

Proof. Let C_{T} (V) ≤ Y ⊴ X with Y^{∗} ≠ 1. Suppose for the moment that Aut_{Y} (P) ≤ Aut_{B}(P). Then by 6.5.2, X′ = N_{X}(T ∩ Y) ∈ Y, so that X′ ∈ X, and hence X′ = X by minimality of X. Therefore T ∩ Y ≤ O_{2}(X) = C_{T} (V) by 7.2.5, so Y^{∗} is of odd order.

Suppose that F^{∗}(X^{∗}) = F (X^{∗}). By 7.2.6, X = J(X)T, so as C_{X^{∗}}(F^{∗}(X^{∗})) ≤ F^{∗}(X^{∗}), we conclude [J(X), F (X^{∗})] ≠ 1. Therefore by B.1.9 in [ASm], J(X)^{∗} = L^{∗} × C_{J(X)}(L^{∗}), where L^{∗} is the direct product of copies of L_{2}(2), and [J(X), F (X^{∗})] = O(L^{∗}). As L^{∗} is not of odd order, it follows from the previous paragraph that Aut_{L}(P) ≰ Aut_{B}(P), so that LT ∈ X, and then X = LT by minimality of X. This contradicts the hypothesis that X is not solvable.

Therefore we can pick a component L^{∗} of X^{∗}, and the preimage Y of ⟨L^{∗X}⟩ is normal in X. As Y^{∗} is not of odd order, it follows from paragraph one that Aut_{Y} (P) ≰ Aut_{B}(P), so that X = Y T. In particular L = L^{∗X} is the set of components of X^{∗}. Let O be an orbit of T on L, and let K be the preimage in X of ⟨O⟩. As X = Y T, K ⊴ X. If Aut_{K}(P) ≤ Aut_{B}(P) then by 6.5.2, N_{X}(T ∩ K) ∈ X, contrary to the minimality of X. Therefore Aut_{K}(P) ≰ Aut_{B}(P), so T K ∈ X, and hence X = T K, so that O = L, completing the proof of (1).

Part (2) follows from B.2.7 and B.1.5.4 in [ASm]. By (2), L_{i}^{∗} = [L_{i}^{∗}, A] for some A ∈ A(T), so by B.2.7 and B.1.11 in [ASm], L_{i}^{∗} is the generalized Fitting subgroup of some almost simple group with an F-module. Then Theorem B in [GM] implies (3).

Assume the setup of (4), and let j ∈ {1, 2}. If Aut_{Y_{j}}(P) ≤ Aut_{B}(P), then by 6.5.2, T ∩ Y_{j} ≤ P and N_{X}(T ∩ Y_{j}) ∈ Y. Then (4) holds in this case with i = 3 − j. So assume Aut_{Y_{j}}(P) ≰ Aut_{B}(P). Then Y_{j}P (T ∩ Y_{3−j}) ∈ Y, so T ∩ Y_{3−j} ≤ P by 6.5.2. Hence (4) holds in this case with i = j.

By 6.4.2, P = O_{2}(N_{X}(P)), so P^{∗} = O_{2}(N_{X^{∗}}(P^{∗})) by 7.3.1. Then (5) follows from C.2.2 in [ASm].

Part (6) follows from (4) and an easy induction argument.

Hypothesis 7.5. Assume Hypothesis 7.1 and X is not solvable. Let I = {1, …, r} and L = {L_{i} : i ∈ I} the set of preimages in X of components of X^{∗}. For i ∈ I, set P_{i} = P ∩L_{i}, T_{i} = T ∩L_{i}, P (i) = N_{P} (L_{i}), T (i) = N_{T} (L_{i}), K_{i} = N_{L_{i}}(P_{i}), H_{i} = N_{L_{i}}(P (i)), and K̄_{i} = K_{i}^{∗}/P_{i}^{∗}. Pick the orbit O of P on L as in 7.4.6, and let I_{0} = {i ∈ I : L_{i} ∈ O}. Choose notation so that 1 ∈ I_{0}.

(7.6) Assume Hypothesis 7.5. Then (1) For i ∈ I, H_{i} ≤ K_{i} and H̄_{i} = CK̄_{i}(P (i)).

(2) H̄_{i} has a strongly embedded subgroup.

(3) K_{1}^{∗} is not contained in any proper T (1)-invariant subgroup of L_{1}^{∗}.

(4) There exists a faithful FF-module for Aut_{T^{∗}}(L_{1}^{∗})L_{1}^{∗}.

(5) If L_{1}^{∗} ≅ SL_{3}(2^{e}), Sp_{4}(2^{e}), A_{6}, or Â_{6}, then T (1)^{∗} is trivial on the Dynkin diagram of L_{1}^{∗}.

Proof. As L_{i} ⊴ L_{i}P (i), P_{i} ⊴ H_{i}, so H_{i} ≤ K_{i}. As P_{i}^{∗} = P (i)^{∗} ∩ K_{i}^{∗}, H̄_{i} = CK̄_{i}(P (i)). Thus (1) holds.

Let

L = ∏_{i∈I_{0}} L_{i}, K = ∏_{i∈I_{0}} K_{i}, P_{0} = ∏_{i∈I_{0}} P_{i}, H_{0} = ∏_{i∈I_{0}} H_{i}, and K̄ = K/P_{0}.

Then K̄ is the direct product of the groups K̄_{i}, i ∈ I_{0}, and by 7.4.5, P_{0} = P ∩ L. From 7.5, O is an orbit of P on L, so P is transitive on {J_{i} : i ∈ I_{0}} for J ∈ {L, K, P }. By (1), H̄_{1} = CK̄_{1}(P (1)), so CK̄(P) is a full diagonal subgroup of the direct product H̄_{0} with CK̄(P) ≅ H̄_{1}.

Let H be the preimage in K of CK̄(P). Then H = N_{L}(P), and by choice of O in 7.5 and 7.4.6, Aut_{L}(P) ≠ Aut_{B}(P). Thus LP ∈ Y, so by 6.4.2, Out_{B}(P) ∩ Out_{L}(P) is strongly embedded in Out_{L}(P). But Out_{L}(P)/Out_{C_{X}(V)}(P) ≅ H̄, and by 7.2.5, Out_{C_{X}(V)}(P) is of odd order and contained in Out_{B}(P). Therefore H̄ has a strongly embedded subgroup. Then as H̄ = CK̄(P) ≅ H̄_{1}, (2) holds.

Suppose K_{1}^{∗} < Y_{1}^{∗} < L_{1}^{∗} with T (1) ≤ N_{T} (Y_{1}^{∗}). Let Y_{1} be the preimage of Y_{1}^{∗} in X and Y = ⟨Y_{1}^{T} ⟩. As Y_{1}^{∗} < L_{1}^{∗}, Y is properly contained in the preimage E of E(X^{∗}) in X, so Y T < X. On the other hand N_{E}(P) ≤ N_{E}(P_{1}). Further N_{L}(P) ≤ ⟨N_{L_{1}}(P_{1})^{P} ⟩ ≤ Y, so as Aut_{L}(P) ≰ Aut_{B}(P), Aut_{Y} (P) ≰ Aut_{B}(P), and hence Y T ∈ X, contrary to the minimality of X. This proves (3).

We next prove (4), using the argument in B.1.11 of [ASm]. By 7.4.2, L_{1}^{∗} ⊴ J(X)^{∗}, and by 7.4.1, T is transitive on L, so there exists A^{∗} ∈ P(T^{∗}, V) with L_{1}^{∗} = [L_{1}^{∗}, A^{∗}]. Let G_{1} = L_{1}^{∗}T (1)^{∗}. Arguing as in the proof of B.1.11 of [ASm], we obtain a G_{1}-submodule U of V with Q = C_{G_{1}}(U) = O_{2}(G_{1}). Then B = L_{1}^{∗}Aut_{T^{∗}}(L_{1}^{∗}) ≅ G_{1}/Q and U is a faithful F_{1}B-module. By B.1.5.1 in [ASm], Aut_{A^{∗}}(U) ∈ P(B, U), so U is an FF-module. This establishes (4).

Finally (5) follows from (4) and B.5.1.1 in [ASm].

(7.7) Assume Hypothesis 7.5 with L_{1}^{∗} of Lie type and characteristic 2. Then L_{1}^{∗} ≅ L_{2}(2^{e}) for some e > 1, and P_{1}^{∗} = 1.

Proof. Suppose first that L_{1}^{∗} ≅ L_{2}(2^{e}). If P_{1}^{∗} ≠ 1 then P_{1}^{∗} = T_{1}^{∗} by 7.4.5. But then the Borel subgroup K_{1}^{∗} of L_{1}^{∗} is a proper T (1)-invariant subgroup of L_{1}^{∗}, contrary to 7.6.3. Hence P_{1}^{∗} = 1, so the lemma holds in this case.

Therefore we may assume L_{1}^{∗} is not L_{2}(2^{e}). However by inspection of the groups listed in 7.4.3, L_{2}(2^{e}) is the only rank 1 group of Lie type on that list, so:

(a) The Lie rank l of L_{1}^{∗} is greater than 1.

Let Δ = {1, …, l} be the Dynkin diagram of L_{1}^{∗}. By 7.4.5, K_{1}^{∗} is a parabolic subgroup of L_{1}^{∗} with unipotent radical P_{1}^{∗}. In particular (cf. I.2.5 in [ASm]) P_{1}^{∗} is weakly closed with respect to L_{1}^{∗} in each 2-subgroup of L_{1}^{∗} containing P_{1}^{∗}, so P_{1}^{∗} ⊴ T_{1}^{∗}. Thus:

(b) K_{1}^{∗} is a parabolic subgroup of L_{1}^{∗} containing T_{1}^{∗}, and P_{1}^{∗} is the radical of K_{1}^{∗}. Hence K_{1}^{∗} = P_{J} for some J ⊆ Δ, in the notation of section 43 of [FGT].

(c) K_{1}^{∗} ≠ L_{1}^{∗}, so J ≠ Δ.

Suppose otherwise. Then K̄_{1} = L_{1}^{∗} and H̄_{1} = H_{1}^{∗}. By 7.6.1, H_{1}^{∗} = C_{L_{1}^{∗}}(P (1)^{∗}). Further by 7.6.2, H_{1}^{∗} has a strongly embedded subgroup. Therefore by 7.4.3 and 4.3, one of the following holds:

(i) L_{1}^{∗} is L_{2}(2^{e}) or U_{3}(2^{e}), or (ii) L_{1}^{∗} is SL_{3}(2^{e}), Sp_{4}(2^{e}), or A_{6} = Sp_{6}(2)′, and P (1)^{∗} is nontrivial on Δ. Case (i) does not hold by (a), while case (ii) does not hold by 7.6.5. This completes

the proof of (c).

Next by 7.6.3, K_{1}^{∗} is not contained in any proper T (1)-invariant subgroup of L_{1}^{∗}. Hence from (b), T (1)^{∗} does not act on P_{J′} for any proper parabolic P_{J′} containing P_{J}; that is, in its action on Δ:

(d) T (1)^{∗} does not act on any proper subset J′ of Δ containing J. In particular by (c) and (d), T (1)^{∗} is nontrivial on Δ. Hence by 7.6.5, L_{1}^{∗} is not

SL_{3}(2^{e}), Sp_{4}(2^{e}), or A_{6} ≅ Sp_{4}(2)′. Moreover by inspection of the list in 7.4.3, the only remaining groups of Lie rank 2 on that list are U_{4}(q), U_{5}(q), and G_{2}(q). But in each of these cases, Aut(L_{1}^{∗}) is trivial on Δ, so we conclude:

(e) l > 2. Next T (1)^{∗Δ} is of order 2, so if P (1)^{∗Δ} ≠ 1 then T (1)^{∗Δ} = P (1)^{∗Δ}. Hence as P (1)^{∗}

acts on P_{J}, and hence also on J, T (1)^{∗} acts on J, contrary to (c) and (d). Thus: (f) P (1)^{∗} is trivial on Δ. Next K̄_{1} = K_{1}^{∗}/P_{1}^{∗} is isomorphic to a Levi factor of the parabolic K_{1}^{∗}. By 7.4.3 and

(e), L_{1}^{∗} is a classical group, so the Levi factor K̄_{1} is a product of classical groups. Further by (f), P (1)^{∗} acts on each of the classical factors of K̄_{1}, and is trivial on the Dynkin diagram of each of those factors. But by 7.6.1, H̄_{1} = CK̄_{1}(P (1)^{∗}), while by 7.6.2, H̄_{1} has a strongly embedded subgroup. It follows that K̄_{1} has just one classical factor Ȳ_{1}, and then by 4.3 either:

(I) Ȳ_{1} ≅ L_{2}(2^{e}) or U_{3}(2^{e}), or (II) Ȳ_{1} ≅ SL_{3}(2^{e}) or Sp_{4}(2^{e}) and P (1)^{∗} is nontrivial on the diagram of Ȳ_{1}. We just observed that P (1)^{∗} is trivial on the diagram of Ȳ_{1}, so (II) does not hold.

Therefore Ȳ_{1} is of Lie rank 1, so K_{1}^{∗} is a minimal parabolic, and hence:

(g) |J| = 1. Recall the orbits of T (1)^{∗} on Δ are of length at most 2. It follows from this fact, (g),

and (d) that:

(h) l = |Δ| = 2. But now (e) and (h) supply a contradiction, completing the proof of the lemma.

(7.8) Assume Hypothesis 7.5. Then L_{1}^{∗} is not Â_{6}.

Proof. Assume otherwise. By 7.6.5, T (1)^{∗} is trivial on the diagram of L_{1}^{∗}, so Σ = Aut_{T} (L_{1}^{∗})L_{1}^{∗} ≅ Â_{6} or Ŝ_{6}. In either case each member of B_{∗}(Σ) contains a Sylow 2subgroup of Σ, so 7.6.3 supplies a contradiction.

(7.9) Assume Hypothesis 7.5 with L_{1}^{∗} ≅ A_{n} an alternating group. Then n = 2^{e} + 1 for some integer e ≥ 2.

Proof. By 7.7, n ≠ 6, so Σ = Aut_{P} (L_{1}^{∗})L_{1}^{∗} ≅ A_{n} or S_{n}. By 7.6.2, Δ = Aut_{P} (L_{1}^{∗}) ∈ B_{∗}(Σ), while by 7.6.3, Σ = ⟨N_{Σ}(Δ), Aut_{T} (L_{1}^{∗})⟩. Then the lemma follows from Theorem 5.12.

Recall the definition of the sets 𝔐(F), 𝔐_{−}(F), 𝔐_{Ω,−}(F), and 𝔐_{+}(F) of minimal parabolics of members of 𝔊(F) in Definition 2.4.

(7.10) Either X ∈ 𝔐_{+}(F) or X ∈ 𝔐_{Ω,−}(F).

Proof. First suppose that X is a minimal parabolic which satisfies condition (1) in Definition 2.4. If S(X) ≠ S, then X satisfies condition (3) of Definition 2.4 by 7.2.2. Moreover condition (3) implies condition (2), so in this case X ∈ 𝔐_{−}(F). On the other hand if S(X) = S then as X ∈ X and B contains C_{F}(Z) and N_{F}(Baum(S)), neither Z nor Baum(S) is normal in X, so condition (2) is satisfied and X ∈ 𝔐_{+}(F). By 7.2.2, O_{2}(X) ∈ W(P), so O_{2}(X) ∈ F_{Ω}^{f} by 6.4.1. Therefore if X ∈ 𝔐_{−}(F) then X ∈ 𝔐_{Ω,−}(X). Thus it remains to show that X is a minimal parabolic and satisfies condition (1) of Definition 2.4.

Suppose X is solvable. Then condition (1) of Definition 2.4 is satisfied trivially. By 7.2.4 and B.2.16 in [ASm], J(X)^{∗} = L_{1}^{∗} × ⋯ × L_{n}^{∗} with L_{i}^{∗} ≅ L_{2}(2) and T acting on L = {L_{i}^{∗} : 1 ≤ i ≤ n}. Then an argument in the proof of 7.4.1 shows T is transitive on L. By 7.2.6, X = J(X)T, so T^{∗} is maximal in X^{∗} = J(X)^{∗}T^{∗}, and hence X^{∗} is a minimal parabolic. Therefore X is minimal parabolic by 7.2.3, so the lemma holds when X is solvable. Therefore we may assume Hypothesis 7.5 holds.

By 7.4.3, either L_{1}^{∗} is of Lie type of characteristic 2, or L_{1}^{∗} ≅ Â_{6}, or L_{1}^{∗} ≅ A_{n} for some n ≥ 5. The second case does not hold by 7.8. In the first case, L_{1}^{∗} ≅ L_{2}(2^{e}) for some integer e ≥ 2 by 7.7. In the third case, L_{1}^{∗} ≅ A_{2^{e}+1} for some e ≥ 2 by 7.9. In particular in each case, Aut_{T} (L_{1}^{∗})L_{1}^{∗} is a minimal parabolic over Aut(L_{1}^{∗}). Moreover by 7.4.1, X^{∗} = E(X^{∗})T^{∗} with T^{∗} transitive on the set L = {L_{1}^{∗}, …, L_{r}^{∗}} of components of X^{∗}, so as Aut_{T} (L_{1}^{∗})L_{1}^{∗} is a minimal parabolic, so is X^{∗}. Therefore X is a minimal parabolic by 7.3.3.

Let L ∈ C(X). Recall X^{∗} = X/C_{X}(V) and by 7.2.5, C_{X}(V) is 2-closed, so C_{X}(V) is solvable. But as L ∈ C(X), L is perfect and subnormal in X, with L/O_{∞}(L) simple by A.3.3.1 in [ASm]. It follows that L^{∗} ∈ L, so L^{∗} ≅ L_{2}(2^{e}) or A_{2^{e}+1}, and then that L^{∗} ≅ L/O_{∞}(L). This completes the proof that X is a minimal parabolic satisfying condition (1) of Definition 2.4, and hence also completes the proof of the lemma.

Definition 7.11. Define B_{Ω}(F) to be the subsystem of F generated by C_{F}(Ω_{1}(Z(S))), N_{F}(Baum(S)), and the subsystems F_{S(M)}(M), as M varies over 𝔐_{+}(F) ∪ 𝔐_{Ω,−}(F). Write B(F) for B_{∅}(F).

Theorem 7.12. Assume F is a saturated fusion system on a finite 2-group S. Assume in addition that F is a local CK-system and Ω is an F-invariant collection of subgroups of S such that F is an Ω-system. Then F = B_{Ω}(F).

Proof. Set B = B(F) and assume the theorem fails for F, so that B ≠ F. Then Hypothesis 6.3 is satisfied for any choice of P ∈ U and W ∈ W(P). In particular there exists X ∈ X_{∗} = X_{∗}(W, P).

By Definition 7.11, C_{F}(Z) ≤ B ≥ N_{F}(Baum(S)), so Hypothesis 7.1 is satisfied. Then by 7.10, X ∈ 𝔐_{Ω,+}(F) or X ∈ 𝔐_{Ω,−}(F). Therefore by Definition 7.11, F_{T} (X) ≤ B, impossible as X ∈ X, so by definition in 6.3, Aut_{X}(P) ≰ Aut_{B}(P). This contradiction establishes the theorem.

Corollary 7.13. Assume F is a saturated fusion system on a finite 2-group S. Assume in addition that F is a local CK-system of characteristic 2-type. Then F = B(F).

Proof. As F is of characteristic 2-type, F is an ∅-system, so the corollary is a special case of 7.12.

Section 8. Glauberman-Niles pairs for F

In this section we assume:

Hypothesis 8.1. F is a saturated fusion system on a finite 2-group S. In addition, F is a local CK-system.

Throughout this section, Ω denotes an F-invariant collection of subgroups of S. Sometimes we also operate under the following hypothesis:

Hypothesis 8.2. Hypothesis 8.1 holds and X ∈ 𝔐_{+}(F) ∪ 𝔐_{−}(F). Set T = S(X), T_{0} = Baum(T), Z_{T} = Ω_{1}(Z(T)), V = Ω_{1}(Z(O_{2}(X))), Ṽ = V/C_{V} (J(X)), and X^{∗} = X/C_{X}(V)). Let M be the unique maximal overgroup of T in X and set E = Ω_{1}(Z(T_{0})).

(8.3) Assume Hypothesis 8.2. Then for some positive integer s:

(1) J(X)^{∗} = X_{1}^{∗} × ⋯ × X_{s}^{∗} and [Ṽ , J(X)^{∗}] = Ṽ_{1} ⊕ ⋯ ⊕ Ṽ_{s}, where V_{i} = [V, X_{i}^{∗}] and [V_{i}, X_{j}^{∗}] = 0, for i ≠ j in I = {1, …, s}.

(2) T is transitive on L = {X_{i}^{∗} : i ∈ I}.

(3) J(X) = O^{2}(X)J(T).

(4) Either

(i) X_{i}^{∗} ≅ L_{2}(2^{e}) for some positive integer e, M^{∗} ∩ X_{i}^{∗} = N_{X_{i}^{∗}}(T^{∗} ∩ X_{i}^{∗}), and Ṽ_{i} is the natural module for X_{i}^{∗}, or

(ii) X_{i}^{∗} ≅ S_{2^{f} +1} for some integer f > 1, M^{∗} ∩ X_{i}^{∗} ≅ S_{2^{f}}, and V_{i} is the core of the (2^{f} + 1)-dimensional permutation module of rank 2^{f}.

(5) T_{0}^{∗} acts on each member of L.

(6) In case (4i), T_{0}^{∗} = J(T)^{∗} ∈ Syl_{2}(J(X)^{∗}), T_{0} ∈ Syl_{2}(O^{2}(X)T_{0}), and E = E_{0} ⊕ E_{1} ⊕ ⋯ ⊕ E_{s}, where E_{0} = C_{E}(J(X)), and for i ∈ I, E_{i} = C_{V_{i}}(T^{∗} ∩ X_{i}^{∗}) with m(Ẽ_{i}) = e.

(7) In case (4ii), T_{0}^{∗} = J(T)^{∗} = T_{0,1}^{∗} × ⋯ × T_{0,s}^{∗}, where T_{0,i}^{∗} ≅ E_{2^{2^{f−1}}} is the subgroup generated by the transpositions in T^{∗} ∩ X_{i}^{∗}, and E = E_{0} ⊕ E_{1} ⊕ ⋯ ⊕ E_{s}, where E_{0} = C_{E}(J(X)) and E_{i} = C_{V_{i}}(T_{0,i}^{∗}) is of rank 2^{f−1}. Moreover C_{T_{0}}(V) = O_{2}(O^{2}(X)T_{0}).

(8) In case (4ii), there exists a T_{0}-invariant subgroup K_{i} = O^{2}(K_{i}) ≤ X_{i} such that K_{i} = [K_{i}, J(T)], K_{i}^{∗}T_{0}^{∗} ≅ S_{3}, and K_{i}T_{0} ∈ G(T_{0}).

Proof. Parts (1)-(4) appear as parts (2)-(5) of B.6.9 in [ASm].

Suppose case (4i) holds. Then (5) and (6) hold in this case by B.6.10 in [ASm].

Thus we may assume case (4ii) holds. Then from (4ii) and B.3.2 in [ASm], J(T)^{∗} = T_{0,1}^{∗} ×⋯×T_{0,s}^{∗}, where T_{0,i}^{∗} is the subgroup described in (7). Moreover by B.3.2 in [ASm], E′_{i} = C_{V_{i}}(T_{0,i}) is of rank 2^{f−1}. Let R be the preimage of J(T)^{∗} in T.

Set L = O^{2}(X). We now verify the hypotheses of B.2.18 in [ASm]. By definition of R, hypothesis (a) of that lemma is satisfied. As X is a minimal parabolic, hypothesis (b) is satisfied. By B.6.9.1 in [ASm], hypothesis (c) is satisfied. By (4ii), X_{i}^{∗} acts as Sym(Ω) on Ω = {1, …, 2^{f} + 1} with T_{0,i}^{∗} = ⟨(1, 2), (3, 4), …, (2^{f} − 1, 2^{f})⟩. Then T_{0,i}^{∗} is conjugate to X_{0,i}^{∗} = ⟨(2, 3)(4, 5), …, (2^{f}, 2^{f} + 1)⟩, and X_{i}^{∗} = ⟨T_{0,i}^{∗}, X_{0,i}^{∗}⟩, so hypothesis (d) and (e) are satisfied. This completes the verification of the hypotheses of B.2.18 in [ASm].

We now appeal to B.2.18 in [ASm] to conclude that O_{2}(L) ≤ T_{0}, E = C_{E}(L)C_{V} (J(T)), and C_{T_{0}}(V) ∈ Syl_{2}(C_{LT_{0}}(V)). Then as C_{X}(V) is 2-closed, C_{T_{0}}(V) = O_{2}(LT_{0}). Further E = E_{0}C_{V} (J(T)) = E_{0} ⊕E_{1} ⊕⋯⊕E_{s}, completing the proof of (7). Also T_{0} ≤ C_{T} (E_{i}) ≤ N_{T} (X_{i}^{∗}), so (5) holds in this case, completing the proof of (5).

Let K̄_{i}^{∗} = (X_{i}^{∗})_{Ω−{2^{f} −1,2^{f} ,2^{f} +1}}, K̄_{i} the preimage of K̄_{i}^{∗} in X, and J_{i} = O^{2}(K̄_{i}). From (7), T_{0} acts on J_{i}. By 1.2, J_{i}T_{0} is generated by its minimal parabolics over T_{0}, so for some minimal parabolic P_{i}, P_{i}^{∗} does not act on T_{0}^{∗}. Thus P_{i}^{∗} = J_{i}^{∗}T_{0}^{∗}. Set K_{i} = O^{2}(P_{i}). Then K_{i}T_{0} ∈ G(T_{0}), completing the proof of (8) and the lemma.

Notation 8.4. Suppose Hypothesis 8.2 is satisfied. If 8.3.4.i holds, set K_{i} = O^{2}(X_{i}). If 8.3.4.ii holds, choose K_{i} as in 8.3.8. Set 𝔎(X) = K_{i}^{S}. Thus if 8.3.4.i holds then 𝔎(X) = {O^{2}(X_{i}) : i ∈ I}, while if 8.3.4.ii holds then by construction of K_{i} in 8.3.8, K_{i}^{N_{S}(X_{i})} consists of the 2^{f−1} subgroups K of X_{i} such that K^{∗} ≅ Z_{3} moves exactly three points of the set Ω = {1, …, 2^{f} + 1} with X_{i}^{∗} = Sym(Ω), one of those points is the fixed point 2^{f} + 1 for S^{∗} ∩ X_{i}^{∗}, and the remaining two points are a cycle in one of the 2^{f−1} transpositions in S^{∗} ∩ X_{i}^{∗}. Thus in this case 𝔎(X) has 2^{f−1}s members.

Observe that by parts (6) and (8) of 8.3, for each K ∈ 𝔎(X), KT_{0} ∈ G(T_{0}), where G(T_{0}) is defined in Definition 1.3.

Definition 8.5. For P ≤ Q ≤ S, define P to be F-characteristic in Q if Aut_{F}(Q) ≤ N_{Aut(Q)}(P).

Define a Baumann pair for F to be a pair (C_{1}, C_{2}) of F-characteristic subgroups of S with C_{1} ≤ Ω_{1}(Z(S)) and C_{2} F-characteristic in Baum(S).

Given a Baumann pair (C_{1}, C_{2}) for F, define 𝔐(F, C_{1}, C_{2}) to consist of those M ∈ 𝔐_{+} such that neither C_{1} nor C_{2} is normal in M. Write B_{Ω}(F, C_{1}, C_{2}) for the subsystem of F generated by C_{F}(C_{1}), N_{F}(C_{2}), and the systems F_{S}(M), as M varies over 𝔐(F, C_{1}, C_{2})∪ 𝔐_{Ω,−}(F). Set B(F, C_{1}, C_{2}) = B_{∅}(F, C_{1}, C_{2}).

Example 8.6. Observe that the pair (Ω_{1}(Z(S)), Baum(S)) is the “largest” Baumann pair for F, that 𝔐_{+} = 𝔐(F, Ω_{1}(Z(S)), Baum(S)), and that B(F, Ω_{1}(Z(S)), Baum(S)) = B(F).

Theorem 8.7. Assume Hypothesis 8.1 and let (C_{1}, C_{2}) be a Baumann pair for F. Assume F is an Ω-system. Then F = B_{Ω}(F, C_{1}, C_{2}).

Proof. By Theorem 7.12, F = B_{Ω}(F). Further if M ∈ 𝔐_{+}(F) and C_{i} ⊴ M for i = 1 or 2, then F_{S}(M) is contained in C_{F}(C_{1}) or N_{F}(C_{2}), so F_{S}(M) ⊆ B_{Ω}(F, C_{1}, C_{2}).

Definition 8.8. Define G(F) to be the set of H ∈ G(Baum(S)) such that O^{2}(H) ∈ 𝔎(X) for some X ∈ 𝔐_{+}(F).

Define a Glauberman-Niles pair (or GN-pair) for F to be a Baumann pair (C_{1}, C_{2}) for F such that for each H ∈ G(F), one of the three conclusions of Definition 1.5 holds.

For H ∈ G(F) the Campbell pair for F defined by H is

C_{1}(H) = Ω_{1}(Z(S)) ∩ ⟨C_{V} ((T_{0})^{Aut_{F} (T_{0})}⟩ and C_{2}(H) = ⟨F^{Aut_{F} (T_{0})}⟩,

where T_{0} = Baum(S) and (cf. 1.3) τ(H) = (H, F, V).

Example 8.9. Define F to be S_{3}-free if for each U ∈ F^{f}, Aut_{F}(U) has no section isomorphic to S_{3}.

Suppose F is S_{3}-free. Then 𝔐(F) and G(F) are empty, so (Ω_{1}(Z(S)), Baum(S)) is a GN-pair for F. Then if in addition F is of characteristic 2-type, then by Theorem 8.7,

F = B(F, Ω_{1}(Z(S)), Baum(S)) = ⟨C_{F}(Ω_{1}(Z(S))), N_{F}(Baum(S))⟩.

(8.10) (1) If G(F) is empty then (Ω_{1}(Z(S)), Baum(S)) is a GN-pair for F.

(2) If H ∈ G(F) then the Campbell pair (C_{1}(H), C_{2}(H)) is a GN-pair for F. Proof. By 8.5, (Ω_{1}(Z(S)), Baum(S)) is a Baumann pair for F. Then (1) follows from the definition of a GN-pair for F in 8.8. The proof of (2) is the same as that of 1.12.

(8.11) Assume (C_{1}, C_{2}) is a GN-pair for F and X ∈ 𝔐(F, C_{1}, C_{2}). Then Hypothesis 8.2 is satisfied, so X is described in 8.3. Adopt the notation of 8.3 and Notation 8.4. For i ∈ I, let X_{i} be the preimage of X_{i}^{∗} in X, L_{i} = O^{2}(X_{i}), L = O^{2}(X), and C = {L_{i} : i ∈ I}. Then

(1) L is the central product of the members of C.

(2) T is transitive on C.

(3) For i ∈ I, L_{i} is an A_{3}, L_{2}(2^{e}), or A_{2^{f} +1} block.

(4) For i ∈ I, L_{i} = [L_{i}, J(T)] is T_{0}-invariant.

(5) For each K ∈ 𝔎(X), KT_{0} ∈ G(F), neither C_{1} nor C_{2} is normal in KT_{0}, K is an L_{2}(2^{e})-block if X_{i}^{∗} ≅ L_{2}(2^{e}) with e > 1, and K is an A_{3}-block otherwise. Proof. This is essentially Proposition C.1.26 in [ASm]. The SQKT hypothesis in that lemma is used only to restrict the type of alternating blocks which appear, so when X_{i}^{∗} satisfies 8.3.4.i, the proof of C.1.26 can be repeated essentially verbatim.

So assume that 8.3.4.ii holds. In this case, using Notation 8.4, by 8.3.8 we can pick K ∈ 𝔎(X) with K ≤ X_{i} and KT_{0} ∈ G(T_{0}). If for some j ∈ {1, 2}, C_{j} is normal in KT_{0} then as C_{j} ⊴ S and X is a minimal parabolic, also C_{j} ⊴ ⟨K, S⟩ = X, contrary to the choice of X ∈ 𝔐(C_{1}, C_{2}, F). Therefore neither C_{1} nor C_{2} is normal in KT_{0}, so from Definition 8.8, K = O^{2}(KT_{0}) is an A_{3}-block. This establishes (5). Then using (5) as in the proof of C.1.26 in the case X_{i}^{∗} ≅ S_{5}, L_{i} is an A_{2^{f} +1} block. Then the remaining parts of the lemma follow.

(8.12) Assume X ∈ 𝔐_{−}(F) and let T = S(X). Then for each GN-pair C_{1}, C_{2} for F_{T} (X) such that C_{i} is F-characteristic in T, X ∈ 𝔐(F_{S}(X), C_{1}, C_{2}).

Proof. As X ∈ 𝔐_{−}(F), T ≠ S. Then as C_{i} is F-characteristic in T, T ≠ N_{S}(C_{i}), so as X ∈ 𝔐_{−}(F), C_{i} is not normal in F.

(8.13) Assume H_{1} ∈ G(F) and set C_{j} = C_{j}(H_{1}) for j = 1, 2 and T_{0} = Baum(S). Then either

(1) 𝔐(F, C_{1}, C_{2}) = ∅, and hence if F is of characteristic 2-type and 𝔐_{−}(F) is empty, then F = ⟨C_{F}(C_{1}), N_{F}(C_{2})⟩, or

(2) there exists X_{2} ∈ 𝔐(F, C_{1}, C_{2}), and for K ∈ 𝔎(X_{2}), H_{2} = KT_{0} ∈ G(F) and there exists α ∈ Aut_{F}(T_{0}) such that [V_{1}α, V_{2}] ≠ 1, where τ(H_{i}) = (H_{i}, F_{i}, V_{i}).

Proof. By 8.10.2, (C_{1}, C_{2}) is a GN-pair for F. If F is of characteristic 2-type and both 𝔐_{−}(F) and 𝔐(F, C_{1}, C_{2}) are empty, then by definition of B = B(F, C_{1}, C_{2}) in 8.5, B = ⟨C_{F}(C_{1}), N_{F}(C_{2})⟩. Hence (1) holds in this case by 8.7.

So assume X_{2} ∈ 𝔐(F, C_{1}, C_{2}), and pick K ∈ 𝔎(X_{2}) and set H_{2} = KT_{0}. By 8.11.5, neither C_{1} nor C_{2} is normal in H_{2}, so (2) follows from 1.10.

(8.14) Assume H ∈ G(F) and set C_{j} = C_{j}(H) for j = 1, 2. Assume O^{2}(H) is not a block. Then

(1) 𝔐(F, C_{1}, C_{2}) = ∅.

(2) If 𝔐_{−}(F) = ∅ and F is of characteristic 2-type, then F = ⟨C_{F}(C_{1}), N_{F}(C_{2})⟩.

Proof. By 8.10.2, (C_{1}, C_{2}) is a GN-pair for F. Observe (2) follows from (1) and 8.13, so it remains to establish (1). Suppose X ∈ 𝔐(F, C_{1}, C_{2}) and pick K ∈ 𝔎(X) and set T_{0} = Baum(S) and H_{2} = KT_{0}. By 8.11.5, neither C_{1} nor C_{2} is normal in H_{2}, so by 1.10 there is α ∈ Aut_{F}(T_{0}) with [V_{1}α, V_{2}] ≠ 1, where H_{1} = H and τ(H_{i}) = (H_{i}, F_{i}, V_{i}). Now there exists (H_{α}, F_{α}, V_{α}) ∈ G(T_{0}) with F_{α} = F_{1}α and V_{α} = V_{1}α; indeed we can choose H_{α} to be the model for the fusion system F_{T_{0}}(H)α, so that α̌ : H → H_{α} is an isomorphism. Then by 1.11, O^{2}(H_{α}) is a block, so as α̌ : H → H_{α} is an isomorphism, O^{2}(H) is a block, a contrary to hypothesis.

Section 9. Pushing up

In this section we assume:

Hypothesis 9.1. Hypothesis 8.1 holds and either

(1) Ω is a Baumann-closed collection of subgroups of S, F is an Ω-system, and X ∈ 𝔐_{Ω,−}(F), or

(2) F is of Baumann characteristic 2, and for j ∈ {1, 2}, X̄_{j} ∈ 𝔐_{+}(F), K̄_{j} ∈ 𝔎(X̄_{j}), T_{0} = Baum(S), H̄_{j} = K̄_{j}T_{0}, τ(H̄_{j}) = (H̄_{j}, F̄_{j}, V̄_{j}), and λ ∈ Aut_{F}(T_{0}) with [V̄_{1}, V̄_{2}λ] ≠ 1.

In case (1) set T = S(X) and T_{0} = Baum(T), and pick K_{1} ∈ 𝔎(X). In case (2) set T = S, X = X̄_{1}, and K_{1} = K̄_{1}. Observe X satisfies Hypothesis 8.2, so we adopt the notation of 8.2 and 8.3.

(9.2) Assume case (2) of Hypothesis 9.1 holds.

(1) Let H_{1} = H̄_{1} and let H_{2} be a model for the fusion system F_{T_{0}}(H̄_{2})λ. Then λ : F_{T_{0}}(H̄_{2}) → F_{T_{0}}(H̄_{2})λ is an isomorphism, which induces an isomorphism λ̌ : H̄_{2} → H_{2} extending λ ∈ Aut(T_{0}), H_{i} ∈ G(T_{0}) for i = 1, 2, and τ(H_{i}) = (H_{i}, F_{i}, U_{i}) where F_{1} = F̄_{1}, U_{1} = V̄_{1}, F_{2} = F̄_{2}λ, and U_{2} = V̄_{2}λ.

(2) X ∈ 𝔐(F, C_{1}(H̄_{2}), C_{2}(H̄_{2})).

Proof. Part (1) follows from section 2 of [A5]. As [V̄_{1}, V̄_{2}λ] ≠ 1, C_{2} = C_{2}(H̄_{2}) ≰ O_{2}(H_{1}), so C_{2} is not normal in X. Similarly V̄_{2}λ ≰ Z(H_{1}), by (1) and 1.11.3. Thus C_{2} is not normal in X, completing the proof of (2).

(9.3) Assume case (1) of Hypothesis 9.1 holds and set U_{1} = [O_{2}(K_{1}), K_{1}]. Define χ to consist of those x ∈ N_{S}(T) − T with x^{2} ∈ T and [U_{1}, U_{1}^{x}] ≠ 1. Then χ ≠ ∅.

Proof. In case (1) of Hypothesis 9.1, T ≠ S so there is s ∈ N_{S}(T) − T with s^{2} ∈ T. Set M_{0} = T ⟨s⟩ and U_{0} = ⟨U_{1}^{M_{0}}⟩ and suppose U_{0} ≤ Q_{1} = O_{2}(K_{1}T_{0}). By 8.12 and 8.11.5, K_{1} is a block, so [K_{1}, Q_{1}] = U_{1} ≤ U_{0}, and hence K_{1} ≤ N_{X}(U_{0}). Then as T ≤ M_{0}, Y = ⟨K_{1}, T ⟩ ≤ N_{X}(U_{0}). But as X is a minimal parabolic and K_{1} is not contained in the unique maximal overgroup of T in X, it follows that X = Y ≤ N_{X}(U_{0}). However T < M_{0} ≤ N_{S}(U_{0}), contradicting X ∈ 𝔐_{−}(F).

Therefore U_{0} ≰ Q_{1}, so there exists x ∈ M_{0} with U_{1}^{x} ≰ Q_{1}. Hence as Q_{1} = C_{T_{0}}(U_{1}), [U_{1}, U_{1}^{x}] ≠ 1. Thus x ∈ χ.

Notation 9.4. In case (2) of Hypothesis 9.1 we are given λ ∈ Aut_{F}(T_{0}), and in 9.2 for j ∈ {1, 2} we defined H_{j} and τ(H_{j}) = (H_{j}, F_{j}, U_{j}) ∈ G(T_{0}). In case (1) of Hypothesis 9.1, 9.3 supplies us with x ∈ χ, and we set λ = c_{x} ∈ Aut_{F}(T_{0}) and H_{1} = K_{1}T_{0} ∈ G(T_{0}), τ(H_{1}) = (H_{1}, F_{1}, U_{1}), and as in 9.2, construct the fusion system F_{T_{0}}(H_{1})λ and its model H_{2} ∈ G(T_{0}) with λ̌ : H_{1} → H_{2} an isomorphism extending λ ∈ Aut(T_{0}), and τ(H_{2}) = (H_{2}, F_{2}, U_{2}) with F_{2} = F_{1}λ and U_{2} = U_{1}λ = U_{1}^{x}.

In either case set K_{i} = O^{2}(H_{i}), Q_{i} = O_{2}(H_{i}), D_{i} = C_{T_{0}}(K_{i}), and E_{i} = [U_{i}, T_{0}]. Observe that H_{1} and H_{2} satisfy the hypothesis of Lemma 1.11, and various properities of these subgroups are recorded in that lemma.

(9.5) Let U = Ω_{1}(Z(J(O_{2}(X))). Then U ∈ F^{f}, N_{F}(U) is constrained, and T ∈ Syl_{2}(G(U)).

Proof. Suppose first that 9.1.1 holds. Then F is an Ω-system and as U ⊴ X, it follows from 7.2.2 and 6.4.1 that U ∈ F_{Ω}^{f} and T ∈ Syl_{2}(G(U)). Assume next that 9.1.2 holds. Then T = S, so U ∈ F^{f} and S ∈ Syl_{2}(G(U)). Thus the lemma holds.

(9.6) (1) Φ(D_{1}) = D_{1} ∩ D_{2} = 1.

(2) K_{1} = O^{2}(X), J(X)^{∗} = H_{1}^{∗} ≅ L_{2}(2^{e}) for some positive integer e, and s = 1.

(3) A(T) = A(T_{0}) = {Q_{1}, Q_{2}} and T_{0} = Q_{1}Q_{2}.

(4) X = K_{1}T and H_{1} = J(X).

(5) In case (2) of 9.1, Aut_{F}(T_{0}) acts on Q_{1} and Q_{2}, and we may choose λ = 1, so that S acts on H_{2}, H_{2}/Q_{2} ≅ H_{1}^{∗}, SK_{2} = X̄_{2} ∈ 𝔐_{+}, and K_{2} = O^{2}(SK_{2}).

Proof. Set D_{0} = D_{1} ∩ D_{2} and suppose either Φ(D_{1}) ≠ 1 or D_{0} ≠ 1. Set B = Φ(D_{1}) in the first case, and B = D_{0} in the second. Let α ∈ 𝔄(B) and set B̄ = Bα.

Observe that D_{i} ⊴ T_{0} for i = 1, 2, so B is also normal in T_{0}. In case (2) of Hypothesis 9.1, T_{0} = Baum(S) so T_{0} = T_{0}α ≤ N_{S}(B̄). Then as F is of Baumann characteristic 2, N_{F}(B̄) is constrained. Similarly in case (1) of Hypothesis 9.1, X ∈ 𝔐_{Ω,−}(F), so from Definition 2.4, O_{2}(X) ∈ F_{Ω}^{f} and T = N_{S}(O_{2}(X)). Further by Hypothesis 9.1, Ω is Baumann-closed and so as T_{0} = Baum(T), it follows that T_{0}, and hence also T_{0}α, are in Ω, so that as F is of Ω-characteristic 2, again N_{F}(B̄) is constrained. Set H = G(B̄), Q_{X} = O_{2}(X), U = Ω_{1}(Z(J(Q_{X}))), Ū = Uα, Ū_{1} = U_{1}α, etc. Set S̄ = N_{S}(B̄), H_{C} = C_{H}(B̄), and Q = O_{2}(H_{C}). We first claim:

(a) T̄_{0} = Baum(NS̄(Ū)).

For by 9.5, U ∈ F^{f} and T = N_{S}(U), so there exist β ∈ 𝔄(Ū) with Uαβ = Ūβ = U. Also T_{0} ≤ N_{T} (B) so J(T_{0})αβ = J(T̄_{0})β ≤ NS̄(Ū)β ≤ T. Then as J(T_{0}) = J(T) it follows that J(T̄_{0}) = J(NS̄(Ū)) and J(T̄_{0})β = J(T_{0}). Now Eα = Ē = Ω_{1}(Z(J(T̄_{0}))) and T_{0}αβ = T̄_{0}β ≤ CS̄(Ē)β ≤ C_{T} (E) = T_{0}, so T_{0}α = T̄_{0} = Baum(NS̄(Ū)), establishing (a).

Replacing B by B^{a} for a ∈ N_{G(U)}(T_{0}) with B^{a} ∈ N_{F}(U)^{f}, we may assume B ∈ N_{F}(U)^{f}. Then by 2.2, α : N_{N_{F} (U)}(B) → N_{N_{F} (B̄)}(Ū) is an isomorphism, which induces an isomorphism α̌ : N_{G(U)}(B) → N_{H}(Ū); in particular N_{X}(B)α̌ ≤ N_{H}(Ū). Set K̄_{1} = K_{1}α̌ ≤ N_{H}(Ū). By 1.11.3, Ū_{2} ∩ O_{2}(K̄_{1}T̄_{0}) = Ē_{1}, so Ū_{2} ∩ Q ≤ Ē_{1}. As AutH̄_{2}(Q_{2}λ^{−1}) ≤ Aut_{F}(Q_{2}λ^{−1}) and λ ∈ Aut_{F}(T_{0}), Aut_{H_{2}}(Q_{2}) = AutH̄_{2}(Q_{2}λ^{−1})λ^{∗} ≤ Aut_{F}(Q_{2}). Then Σ = Aut_{H_{2}}(Q_{2})α^{∗} ≤ Aut_{H}(Q̄_{2}) is irreducible on Ū_{2}/(Ē_{1} ∩ D̄_{1}), so as Ū_{2} ∩ Q ≤ Ē_{1} < Ū_{2}, we conclude that Ū_{2} ∩ Q ≤ Ē_{1} ∩ D̄_{1}. But from 1.11.3, C_{U_{1}}(U_{2}) ≤ U_{2}, so Ū_{1} ∩ Q ≤ Ū_{2} ∩ Q ≤ Ē_{1} ∩ D̄_{1}, and hence Ū_{1} ∩ Q = Ū_{1} ∩ D̄_{1} = CŪ_{1}(K̄_{1}).

Let R = D̄_{1}C_{Q}(K̄_{1}). Now K̄_{1} ≤ H_{C} and F^{∗}(H_{C}) = Q, so K̄_{1} is nontrivial on Q. Therefore by the Thompson A × B-Lemma, K̄_{1} is nontrivial on P = C_{Q}(R). Let P_{0} = C_{P} (K̄_{1}) and P_{1}/P_{0} a minimal K̄_{1}-invariant subgroup of N_{P} (P_{0}). If [K̄_{1}, P_{1}] ≤ P_{0} then as K̄_{1} = O^{2}(K̄_{1}), P_{1} ≤ C_{P} (K̄_{1}) = P_{0}, contrary to the choice of P_{1}. Therefore K̄_{1} is nontrivial and irreducible on P_{1}/P_{0}, so [Ū_{1}, P_{1}] ≤ P_{0}.

Let g ∈ P_{1} and A ∈ A(Q̄_{X}). Then A = Ū_{1}(A∩D̄_{1}) by 1.4.5 and 1.11.2, and D̄_{1} ≤ R ≤ CS̄(P), so A^{g} = Ū_{1}^{g}(Ā ∩ D̄_{1}) with Ū_{1}^{g} ≤ Ū_{1}P_{0} ≤ CS̄(Ū), so A^{g} ≤ J(CT̄_{0}(Ū)) = J(Q̄_{X}) by (a). Therefore P_{1} acts on J(Q̄_{X}), and hence also on Ū = Ω_{1}(Z(J(Q̄_{X}))). Therefore by (a), P_{1} acts on T̄_{0}.

Choose t ∈ T̄_{0} − Q̄_{1}. As K̄_{1} is nontrivial on P_{1}/P_{0}, [P_{1}, t] ≰ P_{0}. As P_{1} acts on T̄_{0}, [P_{1}, t] ≤ T̄_{0}. Then as K̄_{1} is irreducible on P_{1}/P_{0} and acts on T̄_{0}∩Q, P_{1} ≤ T̄_{0}P_{0} ≤ NS̄(K̄_{1}), a contradiction as Ū_{1}/CŪ_{1}(K̄_{1}) is the unique noncentral chief factor for K̄_{1}, and we showed above that Ū_{1} ∩ Q ≤ CŪ_{1}(K̄_{1}).

This contradiction establishes (1). Then (1) implies (2). By (2), T acts on Q_{1}. Further as Φ(D_{1}) = 1, A(T) = A(T_{0}) = {Q_{1}, Q_{2}} and T_{0} = Q_{1}Q_{2} by parts (2) and (3) of 1.11, establishing (3).

By (3), T_{0} = J(T_{0}) = J(T), so by (2) and 8.3.3, H_{1} = K_{1}T_{0} = J(X). Then as X is a minimal parabolic, X = O^{2}(X)T = K_{1}T, completing the proof of (4).

Assume case (2) of 9.1 holds. By (3), Λ = Aut_{F}(T_{0}) permutes {Q_{1}, Q_{2}}, and by (4), S acts on Q_{1}. Thus |Λ : N_{Λ}(Q_{1})| is odd, so Λ acts on Q_{1} and Q_{2}. In particular Q_{i} = Q_{i}λ for i = 1, 2. Observe that as Q_{i} ∈ A(S), F_{i} = Q_{i}. If V̄_{2} ≤ Q_{1} then by 1.8, Q_{1} = Q_{2} or C_{V_{1}}(S) ≤ Z(H̄_{2}). However Q_{1} ≠ Q_{2}, and by 1.11.3, C_{V_{1}}(S) = [Q_{1}, Q_{2}], while λ̌ : H̄_{2} → H_{2} is an isomorphism extending λ by 9.2.1. Hence

[Q_{1}, Q_{2}] = [Q_{1}λ, Q_{2}λ] = [Q_{1}, Q_{2}]λ ≤ Z(H̄_{2})λ̌ = Z(H_{2}),

a contradiction. Thus V̄_{2} ≰ Q_{1}, so we may take λ = 1 and hence U_{2} = V̄_{2} and H_{2} = H̄_{2}. As Φ(D_{2}) = 1 is follows that K_{2} = O^{2}(X̄_{2}), so as in the proof of (4), X̄_{2} = K_{2}S and H_{2} = J(X̄_{2}). By 1.11.1, H_{2}^{∗} ≅ H_{1}^{∗}, completing the proof of (5) and the lemma.

Hypothesis 9.7. Case (2) of Hypothesis 9.1 holds. Set G_{i} = G(Q_{i}), X_{1} = X and X_{2} = X̄_{2}.

(9.8) Assume Hypothesis 9.7. Then

(1) H_{i} = J(X_{i}) ⊴ X_{i} = K_{i}S, and we may choose X_{i} to be a minimal parabolic of G_{i}.

(2) N_{G_{i}}(T_{0}) = G(T_{0}).

(3) O_{2}(G_{i}) = O_{2}(X_{i}) = Q_{i} so [O_{2}(G_{i}), K_{i}] = U_{i}.

(4) Either

(i) K_{i} ⊴ G_{i}, or

(ii) K_{i} is an L_{2}(4)-block, and K_{i} ≤ L_{i} ⊴ G_{i}, with L_{i} an exceptional A_{7}-block.

Proof. By parts (4) and (5) of 9.6, H_{i} = J(X_{i}) ⊴ X_{i}, so Q_{i} = O_{2}(H_{i})) ⊴ X_{i}. Thus (1) holds.

By 9.6.5, Aut_{F}(T_{0}) acts on Q_{i}, so by 2.2.5, N_{G_{i}}(T_{0}) is a model of N_{F}(T_{0}), so (2) holds.

As S ∈ Syl_{2}(G_{i}) and S ≤ X_{i} ≤ G_{i}, O_{2}(G_{i}) ≤ O_{2}(X_{i}), and hence [O_{2}(G_{i}), K_{i}] = U_{i}. Let Y = ⟨K_{i}^{G_{i}}⟩, U_{0} = ⟨U_{i}^{G_{i}}⟩, and Q = O_{2}(G_{i}). Set R = O_{2}(X_{i}). Then as

S ∈ Syl_{2}(G_{i}) acts on K_{i}, R ∈ Syl_{2}(C_{G_{i}}(K_{i}/U_{i})) and R = O_{2}(N_{G_{i}}(R)). Therefore by Theorem 6.4 in [A4], and as S ∈ Syl_{2}(G_{i}) acts on K_{i}, K_{i} ≤ L_{i} ⊴ ⊴ G_{i}, such that one of the following holds:

(a) L_{i} = K_{i}, or

(b) K_{i} is an L_{2}(4)-block and L_{i} is an exceptional A_{7}-block, or

(c) K_{i} is an A_{3}-block and L_{i} is an A_{n}-block for some odd n > 3; indeed K_{i} → L_{i} in the notation of [A4].

Case (c) is impossible as A(S) is of order 2. In cases (a) and (b), L_{i} is a block with O_{2}(L_{i}) = U_{i}. By C.1.9 in [ASm], distinct conjugates of L_{i} commute so Y is a central product of the G_{i}-conjugates of L_{i}, and U_{0} is a central product of the G_{i}-conjugates of U_{i}. Next U_{i} ≤ Q_{i}, and by construction Q_{i} ⊴ G_{i}, so U_{0} ≤ Q_{i}. Thus [U_{0}, T_{0}] ≤ [Q_{i}, T_{0}] = U_{i}. Thus [Y, T_{0}] centralizes U_{0}/U_{i}, so as Y is the central product of the conjugates of L_{i} and U_{0} is the central product of the conjugates of U_{i}, L_{i} = [Y, T_{0}]. Thus T_{0} is in the kernel K of the action of G_{i} on L_{i}^{G_{i}}, so by a Frattini argument, G_{i} = KN_{G_{i}}(T_{0}). Now N_{G_{i}}(T_{0}) acts on L_{i} = [Y, T_{0}] so G_{i} = KN_{G_{i}}(T_{0}) acts on L_{i}, completing the proof of (4).

We’ve seen that O_{2}(G_{i}) ≤ O_{2}(X_{i}) and Q_{i} ⊴ G_{i}. Thus to complete the proof of (3), it remains to show Q_{i} = O_{2}(X_{i}). This can be retrieved from parts of the proof of C.5.6 in [ASm]. Namely an argument on page 159 of [ASm] shows that O_{2}(X_{i}) = C_{S}(U_{i}) = U_{i}C_{S}(K_{i}), and then that C_{S}(K_{i}) ≤ T_{0}, so that O_{2}(X_{i}) = T_{0} ∩ O_{2}(X_{i}) = O_{2}(H_{i}) = Q_{i}.

Hypothesis 9.9. Hypothesis 9.1.1 holds. Set T = S(X), T_{0} = Baum(T), K = O^{2}(X), Q = O_{2}(X), D = C_{T} (K), and U = [K, Q].

(9.10) Assume Hypothesis 9.9 holds. Then (1) K is an A_{3}-block or an L_{2}(2^{e})-block for some e > 1. Set e = 1 if K is an A_{3}-block.

(2) |N_{S}(T) : T | = 2.

(3) J(T) = T_{0} = QQ^{x} for x ∈ N_{S}(T) − T, with A(T) = {Q, Q^{x}}.

(4) Q = UD and m(D) ≤ e.

(5) Either:

(i) |S : T | = 2 and J(T) = J(S). If K is an A_{3}-block then X ≅ S_{4} × Z_{2}.

(ii) K is an A_{3}-block, X ≅ S_{4}, and S is dihedral or semidihedral.

Proof. Part (1) follows from 9.6.2 and 1.11.1. Then by (1) and Hypothesis 9.9, the hypotheses of Theorem C.6.1 in [ASm] are satisfied with Λ = S. Then the lemma follows from that theorem.

Remark 9.11. Let F be a saturated fusion system on a finite p-group S. Define the Alperin data for F to consist of:

(1) S, and

(2) F^{frc}, the set of fully normalized radical centric subgroups of F, and

(3) the subgroups Aut_{F}(R) of Aut(R), as R varies over F^{frc}.

Recall that by Alperin’s Fusion Theorem (cf. A.10 in [BLO]), the Alperin data for F determines F up to isomorphism. To be a bit more precise, suppose F̃ is a saturated fusion system on S̃, and suppose α : S → S̃ is an isomorphism of groups, which induces a bijection α : F^{frc} → F̃^{frc}, such that for each R ∈ F^{frc}, α^{∗} : Aut(R) → Aut(Rα) maps Aut_{F}(R) to AutF̃(Rα). By Alperin’s Fusion Theorem, F = ⟨Aut_{F}(R) : R ∈ F^{frc}⟩ and F̃ = ⟨AutF̃(R̃) : R̃ ∈ F̃^{frc}⟩, so as α preserves the Alperin data, it is immediate that α induces an isomorphism of F with F̃.

(9.12) Assume Hypothesis 9.9 with X ≅ S_{4}. Then |S| = 2^{a} for some a ≥ 4, and F ≅ F_{S}(G) for a finite group G with S ∈ Syl_{2}(G) satisfying one of the following:

(1) S is dihedral and G ≅ L_{2}(q) with 2^{a} = ((q^{2} − 1)/2)_{2}.

(2) S is dihedral and G ≅ P GL_{2}(q) with 2^{a} = (q^{2} − 1)_{2}.

(3) S is semidihedral and G ≅ L_{2}(q)^{(1)}, where q is a square, 2^{a} = (q^{2} − 1)_{2}, and L_{2}(q)^{(1)} is the extension of L_{2}(q) by Z_{2} with semidihedral Sylow 2-subgroups.

(4) S is semidihedral of order 16 and G = L_{3}(3).

Proof. As |S| > |T | = 8, |S| = 2^{a} for some a ≥ 4. By 9.10.5, S is dihedral or semidihedral. By Remark 9.11, it suffices to show that there are four possible isomorphism types of Alperin data, and that the groups listed have such Alperin data.

Observe first that Aut(S) is a 2-group, so as F is saturated, Aut_{F}(S) = Inn(S) is determined.

Suppose that S is dihedral. Then S has two classes Q_{i}^{S}, i ∈ {1, 2}, of 4-subgroups, and we may take Q_{1} = Q. If Aut_{F}(Q_{i}) = Aut_{F}(Q_{i}) ≅ S_{3} for i = 1 and 2, then F^{frc} = Q_{1}^{S} ∪ Q_{2}^{S} ∪ {S}, so the Alperin data is determined in this case, and the groups in (1) have such data. Thus we may assume Aut_{F}(Q_{2}) = Aut_{S}(Q_{2}). In this case F^{frc} = Q^{S} ∪ {S}, so again the Alperin data is determined, and the groups in (2) have such data.

Thus we may assume F is semidihedral. Then S has one class Q^{S} of 4-groups, and X = G(Q) so Aut_{F}(Q) = Aut(Q). Let Z = Z(S) and H = G(Z). Now S has a quaternion subgroup P of index 2, and by an elementary transfer argument, P ∈ Syl_{2}(L) for a normal subgroup L of H. As F^{∗}(H) = O_{2}(H), also F^{∗}(L) = O_{2}(L). Then by another transfer argument, either O^{2}(L) = 1 or a = 4, O^{2}(L) ≅ SL_{2}(3), and H ≅ GL_{2}(3). In the first case, F^{frc} = {S} ∪ Q^{S}, so the Alperin data is determined, and is realized in the groups in case (3). Finally in the second case, F^{frc} = {P, S} ∪ Q^{S}, so the Alperin data is determined, and is realized in L_{3}(3).

(9.13) Assume Hypothesis 9.9 with X ≅ S_{4} × Z_{2}. Set Φ(T) = ⟨z⟩ = Z, let u ∈ U − Z, and D = ⟨d⟩. Then

(1) |S : T | = 2 and there exists an involution x ∈ S − T.

(2) ⟨x, u⟩ ≅ D_{16} and [d, x] = z.

(3) S is a Sylow 2-subgroup of Aut(A_{6}) and Aut(L_{3}(3)).

(4) Z_{2}(S) = ⟨d, uu^{x}⟩ and Z_{2}(S) = C_{S}(Z_{2}(S)).

(5) C_{S}(d) = T and C_{S}(uu^{x}) = ⟨ux, d⟩ ≅ M_{16} the modular group of order 16.

(6) C_{S}(uu^{x}d) = ⟨x, d, uu^{x}⟩ ≅ Z_{4} ∗ Q_{8}.

(7) F^{frc} = {Q, Q^{x}, S} or {Q, Q^{x}, C_{S}(uu^{x}d), S}.

(8) X = G(Q) ≅ G(Q^{x}) and G(S) = S.

(9) If R_{0} = C_{S}(uu^{x}d) ∈ F^{frc} then G(R_{0}) is the unique split extension of R_{0} by Y ≅ S_{3} with an involution y ∈ Y inverting uu^{x}d.

Proof. By 9.10.5, |S : T | = 2. Now Z(T) = ⟨z, d⟩ with [x, d] = z, so Z = Z(S) and the first statement in (5) holds.

Let H = G(Z) and H̄ = H/Z. Then T̄ ≅ E_{8} with CT̄(x̄) = ⟨ūū^{x}, d̄⟩, so (4) follows. Observe also that Φ(S̄) = ⟨ūū^{x}⟩, so Φ(S) = ⟨uu^{x}⟩ is cyclic.

Set P = Z_{2}(S) and y = x^{2}; then ȳ ∈ CT̄(x̄), so y ∈ P. Then y ∈ C_{P} (x) and as [x, d] = z, C_{P} (x) = ⟨uu^{x}⟩ or ⟨uu^{x}d⟩.

Suppose ȳ ∈ ūū^{x}d̄. Then, replacing d by dz if necessary, we may take y = uu^{x}d, so u^{y} = uz. Then

y = y^{x} = (uu^{x}d)^{x} = u^{x}u^{y}d^{x} = u^{x}uzdz = uu^{x}dz ≠ y,

a contradiction. Therefore ȳ ∈ ⟨ūū^{x}⟩. Therefore ⟨x, u⟩ is of order 16 with ⟨u, u^{x}⟩ ≅ D_{8} of index 2, so ⟨x, u⟩ is dihedral or semidihedral of order 16. In the latter case we may take x^{2} = z, and then xd is an involution, so replacing x by xd, (2) holds. Further we have completed the proof of (1).

From (2), we have generators and relations for S, so S is determined up to isomorphism. Let L = Aut(A_{6}) or Aut(L_{3}(3)). Then the fusion system at the prime 2 for L satisfies our hypothesis, so (3) follows from the uniqueness of S. Now (5) and (6) are easy calculation given our generators and relations.

Let R ∈ F^{frc} and G = G(R). Then Z = Z(S) ≤ R. Recall that Φ(S) is cyclic. Thus if Φ(R) ≠ 1 then Z = Ω_{1}(Φ(R)) ⊴ G.

Assume Φ(R) = 1. Now m(R) ≤ m(S) = 3. If m(R) = 3 then by 9.10.3, R ∈ A(S) = {Q, Q^{x}}, so without loss, R = Q. Then G = G(Q). Now T ∈ Syl_{2}(G) with X/Q ≅ S_{3}. Further from the subgroup structure of L_{3}(2), X/Q is maximal in GL(Q) subject to being of order 2n with n odd, so X = G, establishing (8).

Next assume m(R) = 2. Then as C_{S}(R) ≤ R, R is a selfcentralizing 4-subgroup of S, so S is dihedral or semidihedral, contrary to (3).

Therefore we may assume that Φ(R) ≠ 1, so Z ⊴ G, and hence G ≤ H by 2.2.5. As F^{∗}(G) = R, F^{∗}(Ḡ) = R̄. As P̄ ≤ CS̄(R̄) = R̄, P ≤ R. As S̄ centralizes P̄ and R̄ = CS̄(R̄), P < R. Therefore |S : R| ≤ 2, so either R = S or |S : R| = 2, and we may assume the latter. Therefore R = C_{S}(P_{0}) for some subgroup P_{0} of index 2 in P containing Z. Then P_{0} = Z(R) ⊴ G. If P_{0} = ⟨z, d⟩ then 1 < Z < P_{0} < P < R is a G-invariant chain as P is the subgroup of R generated by its elements of order 4. Hence in this case, Aut_{G}(R) is a 2-group, a contradiction. Similarly as Aut(R) is not a 2-group, P_{0} ≠ ⟨uu^{x}⟩ by (5). Therefore P_{0} = ⟨uu^{x}d⟩, so R = P_{0} ∗ B where B ≅ Q_{8} by (6). Thus Out(R) ≅ Z_{2} × S_{3}, so as Aut_{S}(R) ≅ Z_{2} is Sylow in Aut_{G}(R), it follows that Aut_{G}(R) ≅ S_{3}. Let Y_{0} ∈ Syl_{3}(G). By the Baer-Suzuki Theorem, an involution y ∈ S −R inverts some subgroup of order 3, which we may take to be Y_{0}. Then Y = ⟨Y_{0}, y⟩ ≅ S_{3} is a complement to R in G. As R = C_{S}(P_{0}), y inverts P_{0}, completing the proof of (9).

(9.14) Assume Hypothesis 9.9 with X ≅ S_{4} × Z_{2}. Then F ≅ F_{S}(G) for a finite group G with S ∈ Syl_{2}(G) such that either G ≅ Aut(A_{6}) or G ≅ Aut(L_{3}(3)).

Proof. By parts (7)-(9) of 9.13, there are two possible sets of Alperin data for F, so by Remark 9.11, up to isomorphism there are two possibilities for F. By (3), S is Sylow in G_{1} ≅ Aut(A_{6}) and G_{2} ≅ Aut(L_{3}(3)). In the first case, F^{frc} = {Q, Q^{x}, S} and in the second, F^{frc} = {Q, Q^{x}, C_{S}(uu^{x}d), S}, so the lemma follows.

(9.15) Assume Hypothesis 9.7 with K_{1} an A_{3}-block. Then (1) Either H_{1} ≅ H_{2} ≅ S_{4} or H_{1} ≅ H_{2} ≅ S_{4} × Z_{2}.

(2) For i = 1, 2, G_{i} = H_{i}, so S = T_{0}.

(3) If H_{1} ≅ S_{4} then F ≅ F_{S}(G) with S ∈ Syl_{2}(G) and G ≅ L_{3}(2).

(4) If H_{1} ≅ S_{4} × Z_{2} then F ≅ F_{S}(G) where G ≅ S_{6} ≅ Sp_{4}(2).

Proof. By parts (1) and (2) of 1.11, H_{i} = U_{3−i}K_{i} × D_{i} with U_{3−i}K_{i} ≅ S_{4}. By 9.6.1, Φ(D_{i}) = D_{1} ∩ D_{2} = 1, and then by 1.11.6, m(D_{i}) ≤ 1. Therefore (1) holds.

By 9.8.3, O_{2}(G_{i}) = Q_{i}, and by 9.8.4, K_{i} ⊴ G_{i}. Then as F^{∗}(G_{i}) = O_{2}(G_{i}), it follows from (1) that H_{i} = G_{i}, establishing (2).

Assume that H_{1} ≅ S_{4}. By (2), S = T_{0} ≅ D_{8}. Moreover U_{1} and U_{2} are the two 4-subgroups of S, and Aut_{F}(U_{i}) = H_{i}/U_{i} ≅ S_{3}, so Aut_{F}(U_{i}) = Aut(U_{i}). Then, as in the proof of 9.12, (2) holds.

So assume H_{1} ≅ S_{4} × Z_{2}. By (2), S = T_{0} ≅ Z_{2} × D_{8} and G_{i} = H_{i} ≅ Z_{2} × S_{4}.

Let ι = (ι_{i} : S → G_{i}, i = 1, 2) be the amalgam determined by the inclusion maps ι_{i} : S → G_{i}. Visibly ι satisfies Hypothesis A on page 94 of [DGS]; thus, as defined on page 96 of [DGS], ι is a weak BN-pair of rank 2. Therefore by Theorem A on page 100 of [DGS], ι is determined up to isomorphism.

Let R ∈ F^{frc}. Then Z = Z(S) ≤ R and C_{S}(R) ≤ R so Z < R. Then as |Z| = 4 and |S| = 16, R is equal to S or one of the three subgroups Q_{1}, Q_{2}, Q_{3} of S of index 2 containing Z, where Q_{3} ≅ Z_{2} × Z_{4}. However Aut(Q_{3}) is a 2-group, so Q_{3} ∉ F^{frc}. Thus F^{frc} = {S, Q_{1}, Q_{2}}. Then as the amalgam ι is determined up to isomorphism, it follows that the Alperin data for F is determined up to isomorphism, and hence from Remark 9.11, F is determined up to isomorphism. As the fusion system for G = Sp_{4}(2) satisfies our hypothesis, (3) follows.

Section 10. L_{2}(q)-blocks

In this section we assume:

Hypothesis 10.1. Let q = 2^{e} with e > 1 and integer and assume either:

(1) Hypothesis 9.9 holds with K/O_{2}(K) ≅ L_{2}(q), or

(2) Hypothesis 9.7 holds with K_{i}/O_{2}(K_{i}) ≅ L_{2}(q).

Notation 10.2. Set I = {1, 2}. In case (1) of Hypothesis 10.1, let x ∈ S − T and set Q_{1} = Q, Q_{2} = Q^{x}, and for i ∈ I, set G_{i} = G(Q_{i}). By 2.2.5, we may choose notation so that X ≤ G_{1}.

In case (2), Q_{i} and G_{i} = G(Q_{i}) are defined in 9.7 and 9.4. In this case set T = S, X = X_{1}, K = K_{1}, and Q = Q_{1}.

In either case set G_{1,2} = N_{G(T_{0})}(Q). Let F_{i} = N_{F}(Q_{i}), F_{1,2} = N_{N_{F} (T_{0})}(Q), ι_{i} : F_{1,2} → F_{i} the inclusion map, and α_{i} : G_{1,2} → G_{i} the induced map of groups supplied by Lemma 2.5 in [A5]. Let α = (α_{i} : G_{1,2} → G_{i} : i ∈ I) be the corresponding amalgam of groups, (cf. Section 36 in [SG] for a discussion of amalgams and their completions) and G_{+} the universal completion of α, with G_{J} identified with a subgroup of G_{+} for ∅ ≠ J ⊆ I, so that G_{1,2} is a subgroup of G_{1} and G_{2}.

In case (1) of Hypothesis 10.1, observe that c_{x}^{∗} ∈ Aut(F_{1,2}) and c_{x}^{∗} : F_{1} → F_{2} is an isomorphism such that c_{x}^{∗}ι_{2} = ι_{1}c_{x}^{∗}, so c_{x}^{∗} induces a similarity (cf. Definition 11.10) c = (c_{J} : ∅ ≠ J ⊆ I) of α, and then c induces to an automorphism x of G_{+} extending the conjugation map c_{x} ∈ Aut(T). In particular x acts on G_{1,2} and G_{i}^{x} = G_{3−i}. Set K_{1} = K and K_{2} = K_{1}^{x}.

In either case let D_{i} be a Cartan subgroup of K_{i} acting on T_{0}, and in case (1) choose D_{2} = D_{1}^{x}. Set L_{i} = K_{i}Q_{i}. We see in the next lemma that D_{3−i} acts on L_{i} and T_{0}D_{i}; set G̃_{i} = L_{i}D_{3−i} and G̃_{1,2} = T_{0}D_{1}D_{2}, regarded as subgroups of G_{+}. Set G̃_{+} = ⟨G̃_{1}, G̃_{2}⟩ ≤ G_{+} and α̃ = (α_{i} : G̃_{1,2} → G̃_{i} : i ∈ I). We also see that, in case (1) we can choose x so that D_{3−i} acts on D_{i}; make this choice. In case (2) we can of course choose D_{3−i} to act on D_{i}. In either case set B = D_{1}D_{2}.

(10.3) Adopt Notation 10.2. Then (1) For i ∈ I, T ∈ Syl_{2}(G_{i}), and Q_{i} = O_{2}(G_{i}) ∈ A(T).

(2) Either

(a) K_{i} = G_{i}^{∞}, so L_{i} ⊴ G_{i} and D_{3−i} acts on L_{i} and T_{0}D_{i}. Moreover in case (1) of Hypothesis 10.1, we can, and do, choose x so that D_{3−i} acts on D_{i}; or

(b) Case (2) of Hypothesis 10.1 holds, and up to a permutation of I, G_{1} is an exceptional A_{7}-block and G_{2} is the split extension of Q_{2} ≅ E_{16} by ΓL_{2}(4) acting naturally on Q_{2}.

Proof. By 9.5, T ∈ Syl_{2}(G_{i}). In case (2) of 10.1, Q_{i} = O_{2}(G_{i}) by 9.8.3, and Q_{i} ∈ A(T) by 9.6.3. In case (1) of 10.1, Q_{i} ∈ A(T) by 9.10.3, while by definition of Q_{i} in 9.9 and 10.2, Q_{i} = O_{2}(X_{i}) and Q_{i} ⊴ G_{i}. Then as T ≤ X_{i} ≤ G_{i}, it follows that Q_{i} = O_{2}(G_{i}). Hence (1) is established.

Assume case (1) of Hypothesis 10.1 holds. Then the proof of 9.8 shows that either K ⊴ G_{1}, or U = O_{2}(K) ≅ E_{16}, X/Q ≅ S_{5}, and L_{0} = G_{1}^{∞} is an exceptional A_{7}-block with U = O_{2}(L_{0}). Assume the latter. Then Z = C_{U}(T) = Z(T) ≅ Z_{2} and setting L_{Z} = C_{L_{0}}(Z), we have L_{Z}/U ≅ L_{3}(2). Thus x centralizes Z, so S = T ⟨x⟩ centralizes Z. By 2.2, we may take L_{Z} ≤ G(Z). For s ∈ S − T, [QQ^{x}, s] ≰ O_{2}(L_{Z}T), so R = O_{2}(G(Z)) ≤ O_{2}(L_{Z}T) = Q. Then R ≤ Q ∩ Q^{x} = E_{1}D. In particular U = [e, L_{Z}] ≰ R for e ∈ E_{1}D − ZD, so R ≤ ZD ≤ C_{S}(L_{Z}), contradicting R = F^{∗}(G(Z)). Therefore K ⊴ G_{1}.

We have shown that in case (1) of Hypothesis 10.1, for each i ∈ I, K_{i} ⊴ G_{i}. Now remove that assumption that we are in case (1) of 10.1, but assume that for i ∈ I, K_{i} ⊴ G_{i}. Set G_{1}^{∗} = G_{1}/Q. As Q = O_{2}(KQ), C_{G_{1}^{∗}}(K^{∗}) is of odd order, so K^{∗} = G_{1}^{∗∞}, and hence K = G_{1}^{∞}. Now K and Q are normal in G_{1}, so L_{1} = KQ ⊴ G_{1}.

Observe that for i ∈ I, T ∩ L_{i} = T_{0} ∈ Syl_{2}(L_{i}), so N_{L_{i}}(T_{0}) = T_{0}D_{i} ≤ G̃_{1,2} ≤ G̃_{3−i}. Then as L_{1} ⊴ G_{1}, D_{2} acts on L_{1}, and then on N_{L_{1}}(T_{0}) = T_{0}D_{1}. As D_{2} acts on T_{0}D_{1} it acts on some Hall 2′-subgroup of T_{0}D_{1}. Thus in case (1) of 10.1, replacing x by xu for some suitable u ∈ T_{0}, we can assume D_{2} acts on D_{1}. By symmetry D_{1} acts on T_{0}D_{2}, and as D_{1}D_{2} is of odd order, D_{1}D_{2} ∩ T_{0}D_{2} = D_{2}, so D_{1} acts on D_{2}. This completes the proof that (2a) holds when K_{i} ⊴ G_{i}. Moreover we showed that in case (1) of 10.1, it is always the case that K_{i} ⊴ G_{i}. Thus we may assume case (2) of 10.1 holds.

By 9.8.4, either K_{1} ⊴ G_{1} or K_{1} ≤ L_{0} ⊴ G_{1} with L_{0} an exceptional A_{7}-block, and we may assume the latter. As S acts on K_{1}, K_{1}S/Q_{1} ≅ S_{5}. As L_{0} ⊴ G_{1} and L_{0}/Q_{1} is selfnormalizing in GL(Q_{1}), L_{0} = G_{1}. From 9.6.3, A(S) = {Q_{1}, Q_{2}} and T_{0} = Q_{1}Q_{2}. As L_{0} is an exceptional A_{7}-block, we can represent L_{0} on Ω = {1, …, 7} so that Q_{1} is the kernel of the representation and N_{L_{0}}(T_{0}) is the stabilizer in L_{0} of Δ = {1, …, 4}. As G_{1} = L_{0}, G(T_{0}) = N_{G_{1}}(T_{0}) = N_{L_{0}}(T_{0}) by 9.8.2. Also setting Z = Z(S) and L_{Z} = C_{L_{0}}(Z), L_{Z}/Q_{1} ≅ L_{3}(2) and Q_{1} is an indecomposable F_{2}L_{Z}-module with L_{Z} irreducible on Q_{1}/Z. By 2.2.5 we may take L_{Z} ≤ G(Z). Then as S ∈ Syl_{2}(G(Z)) and S ≤ L_{Z}, O_{2}(G(Z)) ≤ O_{2}(L_{Z}) = Q. As Z is the unique proper nontrivial L_{Z}-submodule of Q and F^{∗}(G(Z)) = O_{2}(G(Z)), it follows that Q = O_{2}(G(Z)), so by 2.2, we may take G(Z) ≤ G(Q) = G_{1} = L_{0}, and hence G(Z) = C_{G_{1}}(Z) = L_{Z}.

By symmetry, either K_{2} ⊴ G_{2} or Q_{2} = O_{2}(G(Z)), and as Q_{1} ≠ Q_{2}, the former holds. We saw that G(T_{0}) = N_{L_{0}}(T_{0}) and by 9.8.2, G(T_{0}) = N_{G_{2}}(T_{0}). As N_{L_{0}}(T_{0}) is the stabilizer in L_{0} of Δ, N_{L_{0}}(T_{0}) is the extension of T_{0} by E_{9} inverted by an involution, so as G_{2} = K_{2}N_{G_{2}}(T_{0}), it follows that G_{2} is the split extension of Q_{2} by ΓL_{2}(4). Thus (2b) holds, completing the proof of the lemma.

(10.4) Adopt Notation 10.2 and assume conclusion (2b) of 10.3 holds. Then F = F_{S}(G) where S ∈ Syl_{2}(G) and G ≅ M_{23}.

Proof. Recall the amalgam α = (α_{i} : G_{1,2} → G_{i}, I ∈ I) of 10.2. As conclusion (2b) of 10.3 holds, G_{1} is an exceptional A_{7}-block, G_{2} is the split extension of Q_{2} by ΓL_{2}(4), and G_{1,2} = T_{0}Λ where Λ = N_{G_{1,2}}(Σ) is the extension of Σ ≅ E_{9} by an involution s ∈ S inverting Σ. Recall also from the proof of 10.3 that Z = Z(S) is of order 2 and G(Z) = C_{G_{1}}(Z). We first observe:

(a) F is of characteristic 2-type.

Namely G_{i} is transitive on Q_{i}^{#} for i = 1, 2, and G_{1} is transitive on the involutions in G_{1} −Q_{1}, so F has one class of involutions. In particular Z is the unique member of F^{f} of order 2. By Hypothesis 10.1, F is of Baumann characteristic 2, so C_{F}(Z) is constrained. Now (a) follows from 2.6.

We next observe:

(b) 𝔐_{−}(F) = ∅.

This follows from (a) and Theorem 1, whose proof makes no use of this lemma. We next claim that:

(c) F = ⟨F_{S}(G_{1}), F_{S}(G_{2})⟩.

First K_{1} ∈ G(F), so we can form the Campbell pair (C_{1}, C_{2}) for F, where C_{j} = C_{j}(K_{1}), as in Definition 8.8. Then by (a), (b), and 8.7,

(*)  F = ⟨C_{F}(C_{1}), N_{F}(C_{2}), F_{S}(M) : M ∈ 𝔐(F, C_{1}, C_{2})⟩.

Now as Z is of order 2, C_{1} = Z. Further C_{F}(Z) = F_{S}(G(Z)) ≤ F_{S}(G_{1}) as G(Z) ≤ G_{1}. Similarly C_{2} = Q_{1}, so N_{F}(C_{2}) = F_{S}(G_{1}). Thus we conclude from (*) that:

(**)  F = ⟨F_{S}(G_{1}), F_{S}(M) : M ∈ 𝔐(F, C_{1}, C_{2}).

Further if M ∈ 𝔐(F, C_{1}, C_{2}) then by 8.13, X, M satisfy Hypothesis 9.1.2, so by 9.6.2 and 1.11, O^{2}(M) is an L_{2}(4)-block. Than as A(S) = {Q_{1}, Q_{2}}, it follows that M = X_{2}, completing the proof of (c).

For i ∈ I, let Σ_{i} = C_{Σ}(Q_{3−i}/E_{1}) and observe Σ_{i} ≅ Z_{3} with Σ = Σ_{1} × Σ_{2}. Further N_{S}(Σ_{1}) is a complement in S to Q_{1}, so G_{1} splits over Q_{1} by Gaschutz’ Theorem (cf. 10.4 in [FGT]). Therefore both G_{1} and G_{2}, and hence also G_{1,2} and S are determined up to isomorphism. Suppose F̄ is a fusion system on S satisfying our hypothesis. Then we obtain an amalgam ᾱ = (ᾱ_{i} : Ḡ_{1,2} → Ḡ_{i}, I ∈ I) determined by and determining F̄. Further Ḡ_{J} ≅ G_{J} for ∅ ≠ J ⊆ I, so without loss Ḡ_{J} = G_{J} for J = {1, 2} and {1}, and we have an isomorphism μ : Ḡ_{2} → G_{2} with G_{1,2}μ = G_{1,2} and Q_{2}μ = Q_{2}. We claim:

(d) We can choose μ so that η = μ_{|G_{1,2}} is the identity map.

If so, then F_{S}(G_{i}) = F_{S}(Ḡ_{i}) for i ∈ I, so F = F̄ by (c), and hence F is determined up to isomorphism. We now prove (d). We need γ ∈ Aut(G_{2}) acting on G_{1,2} such that ηγ_{|G_{1,2}} = 1. Then we replace μ by μγ to obtain the desired map.

As A(S) = {Q_{1}, Q_{2}} and η acts on Q_{2}, η also acts on Q_{1}. Adjusting by an inner automorphism of G_{1,2}, we may assume η acts on Σ. Then η acts on Σ_{i} for i ∈ I, so η acts on P_{i} = C_{Q_{i}}(Σ_{i}). Also Σ has two more subgroups Σ_{3} and Σ_{4} of order 3, where Σ_{3} = C_{Σ}(E_{1}) and C_{T_{0}}(Σ_{4}) = 1. Then η acts on these two subgroups also, so η_{|Σ} is in the kernel of the action of GL(Σ) on its projective space, so η inverts or centralizes Σ. Replacing η by ηc_{s} if necessary, we may assume η centralizes Σ.

Let G_{1,2}^{∗} = G_{1,2}/E_{1}. Then T_{0}^{∗} = P_{1}^{∗} ⊕ P_{2}^{∗}, so as η acts on P_{i}^{∗} and centralizes Σ, replacing η by ηc_{σ} for suitable σ ∈ Σ, we may assume η centralizes T_{0}^{∗}. Then [η, Λ] ≤ C_{Λ}(T_{0}^{∗}) = 1.

Let ν be the 2′-part of η. As ν centralizes T_{0}^{∗} and E_{1} = Φ(T_{0}), ν centralizes T_{0}, and hence also G_{1,2} = T_{0}Λ. Therefore η is a 2-element. Then as η acts on P_{i} and E_{1}, and Σ is irreducible on each of these spaces, η centralizes P_{1}P_{2}E_{1} = T_{0}. Thus η = 1, completing the proof of (d).

We have shown that F is determined up to isomorphism. Let G = M_{23}. As the fusion system of G satisfies our hypothesis, we may take S ∈ Syl_{2}(G) and then F = F_{S}(G), completing the proof of the lemma.

Section 11. Weak BN-pairs

In this section we assume:

Hypothesis 11.1. Hypothesis 10.1 holds with K_{i} ⊴ G_{i} for each i ∈ I.

(11.2) Adopt Notation 10.2. Then

(1) For i ∈ I, T ∈ Syl_{2}(G_{i}), Q_{i} = O_{2}(G_{i}), and K_{i} = G_{i}^{∞}, so L_{i} ⊴ G_{i} and D_{3−i} acts on L_{i} and T_{0}D_{i}. Moreover in case (1) of Hypothesis 10.1, we can, and do, choose x so that D_{3−i} acts on D_{i}.

(2) (G_{+}, L_{1}, L_{2}, T) satisfies Hypothesis F.1.1 of [ASm].

(3) (G̃_{+}, L_{1}, L_{2}, T_{0}) satisfies Hypothesis F.1.1 of [ASm] with T_{0} ⊴ G̃_{1,2}.

(4) α and α̃ are weak BN-pairs of rank 2 (cf. Definition F.1.7 in [ASm]).

(5) α̃ is the G_{0}-amalgam, where G_{0} is L_{3}(q) or Sp_{4}(q). Further α is an extension (cf. Definition F.4.3 in [ASm]) of α̃, and hence is an M̄-amalgam for some subgroup M̄ of P ΓL_{3}(q) or ΓSp_{4}(q) containing G_{0}, respectively.

(6) K_{i} = L_{i}.

(7) If α̃ is the L_{3}(q)-amalgam then Q_{i} ≅ E_{2^{2e}} is the natural module for L_{i}/Q_{i} and L_{i}B is isomorphic to a maximal parabolic in L_{3}(q).

(8) If α̃ is the Sp_{4}(q)-amalgam, then Z(L_{i}) ≅ E_{2^{e}}, D_{3−i} acts regularly on Z(L_{i})^{#}, and L_{i} is isomorphic to a maximal parabolic in Sp_{4}(q).

(9) T_{0}B ⊴ G_{1,2}S, C_{T_{0}}(B) = 1, and T is the split extension of T_{0} by a cyclic subgroup F of N_{T} (B) acting faithfully as a group of field automorphisms on L_{i}/Q_{i}.

(10) If f is an involution in F then T_{0} is transitive on the involutions in fT_{0}, and C_{L_{i}}(f) is isomorphic to O^{2′}(P), where P is a parabolic in G_{0}(q^{1/2}), where α̃ is of type G_{0}.

Proof. Part (1) is just a restatement of 10.3, given the fact that Hypothesis 11.1 excludes the exceptional case (b) in 10.3.2.

Given (1), the proofs of parts (2) and (3) are now straightforward. For example O_{2}(G_{+}) = O_{2}(G̃_{+}) = 1 as no nontrivial subgroup of T is normal in both G_{1} and G_{2}. For suppose 1 ≠ W ≤ S with W normal in G_{1} and G_{2}. By 2.2.5, we may take G_{i} ≤ G(W) = G for i = 1, 2. Thus R = O_{2}(G) ≤ Q_{i}. Set U_{i} = [K_{i}, Q_{i}]. If [R, K_{i}] ≠ 1 then U_{i} = [R, K_{i}] ≤ R, contradicting U_{i} ≰ Q_{3−i}. Therefore R ≤ C_{T} (K_{1}) ∩ C_{T} (K_{2}), contary to 9.6.1.

Next α̃ is a weak BN-pair by (1) and F.1.9 in [ASm]. Similarly (1) and the proof of F.1.9 in [ASm] shows that α is a weak BN-pair. So (4) holds.

The first statement in (5) follows from (4) and F.1.12 in [ASm]. By (4) and Theorem A on page 100 of [DGS], α is an M̄-extension of α̃, so the second statement of (5) also holds.

The remaining parts of the lemma (except for the first statement in (9)) follow from (5) and the structure of P ΓL_{3}(q) and ΓSp_{4}(q). Alternatively the proofs of (6)-(9) essentially appear in 2.4.8 of [ASm]. Namely by (5), α̃ is the G_{0}-amalgam, where G_{0} is L_{3}(q) or Sp_{4}(q). In particular G_{i} is isomorphic to a maximal parabolic of G_{0}, so the various remarks in (6)-(8) follow, as does the assertion that C_{T_{0}}(B) = 1. As G_{1,2} ≤ G_{i}, G_{1,2} acts on N_{L_{i}}(T_{0}) = T_{0}D_{i} from (1). Hence T_{0}B = T_{0}D_{1}D_{2} ⊴ G_{1,2}. In case (1) of 10.1, by construction in (1), D_{1}^{x} = D_{2}, while x^{2} ∈ T ≤ N_{S}(T_{0}B), so S = T ⟨x⟩ acts on T_{0}B.

In either of the two cases in 10.1, by a Frattini argument, T = T_{0}F, where F = N_{T} (B), and as C_{T_{0}}(B) = 1, F is a complement to T_{0} in T. Thus F ≅ F^{∗}, and we saw C_{G_{1}^{∗}}(K^{∗}) is of odd order, so F^{∗} acts faithfully on K^{∗} with Aut_{F} (K^{∗}) ∩ Inn(K^{∗}) = 1. It follows that F induces a group of field automorphisms on K^{∗}, so in particular F is cyclic. This completes the proof of (9). See 2.4.12 in [ASm] for the proof of (10).

(11.3) Adopt Notation 10.2 and assume case (1) of Hypothesis 10.1 holds and s ∈ S −T with s^{2} ∈ T_{0}. Then ⟨s⟩T_{0} splits over T_{0}, T_{0} is transitive on the involutions in sT_{0}, and choosing s to be an involution, one of the following holds:

(1) α̃ is of type L_{3}(q), Z(T_{0}) = C_{T_{0}}(s), T_{0} is transitive on s[T_{0}, s], and [T_{0}, s] is transitive on sZ(T_{0}). In this case we say that s is a graph involution.

(2) α̃ is of type L_{3}(q) and C_{T_{0}}(s) is a Suzuki 2-group of order q^{3/2} with |Ω_{1}(Z(C_{T_{0}}(s)))| = q^{1/2}. In this case we say s is a graph-field involution.

(3) α̃ is of type Sp_{4}(q) and C_{T_{0}}(s) is a Suzuki 2-group of order q^{2} with |Ω_{1}(Z(C_{T_{0}}(s)))| = q. In this case we say that s is a Suzuki involution.

Proof. This is essentially 2.4.14 in [ASm]. By 11.2.9, s acts on T_{0}B, so by a Frattini argument, there exists s_{0} ∈ sT_{0} ∩N_{S}(B). By 11.2.9, C_{T_{0}}(B) = 1, so ⟨s_{0}⟩ is a complement to T_{0} in T_{0}⟨s⟩. See 2.4.14 in [ASm] for the rest of the proof.

(11.4) Assume α is the M̄-extension of L_{3}(q). Let z be an involution in Z(S), Z_{0} = Z(T_{0}), Y = O^{2}(C_{G_{1,2}}(z)), Ỹ = CG̃_{1,2}(z), and R = O_{2}(G(z)). Then

(1) z ∈ Z_{0} ⊴ G_{1,2}S.

(2) If q > 4 then T_{0} = O_{2}(Ỹ S) and Ỹ is irreducible on Q/Z_{0}.

(3) Assume q = 4. Then either

(i) O^{2}(M̄) = L_{3}(q) and Y = 1, or

(ii) O^{2}(M̄) = P GL_{2}(q), |Y : T_{0}| = 3, and Y is irreducible on Q/Z_{0}. Moreover either O_{2}(Y S) = T_{0}, or case (1) of 10.1 holds and O_{2}(Y S) = T_{0}⟨s⟩, where s is a graph-field involution.

(4) If T_{0} ≤ R then G(z) = Y S is solvable and Z_{0} is an FT I-subgroup of S.

(5) If q > 4 then T_{0} = R.

(6) Assume O^{2}(M̄) = P GL_{3}(4). Then either T_{0} ≤ R, or case (1) of 10.1 holds, R = ⟨s⟩[s, T_{0}] ≅ Q_{8}D_{8} for some graph-field involution s, and Out_{F}(R) ≅ A_{5} or S_{5}. Further if F is of characteristic 2-type then T = T_{0}, M̄ is P GL_{3}(4), S is Sylow in Ĝ = J_{3}, G(z) ≅ CĜ(z), and G(z)/R ≅ A_{5}.

(7) If O^{2}(M̄) = L_{3}(4) then G(Z_{0}) = G_{1,2}S, T_{0} ≤ R, and G(z) = S.

(8) Assume q = 4. Then T = T_{0} if F is of characteristic 2-type, while in case (2) of Hypothesis 10.1, for each P ∈ F^{frc}, G(P) ≤ G_{i} for i = 1 or 2.

Proof. First Z(T) ≤ Z_{0} by 11.2.5. Further in case (1) of 10.1, for x ∈ S − T, Q ≠ Q^{x}, so x ∉ Z(S). As Z_{0} = Z(T_{0}) and T_{0} = J(S), S acts on Z_{0}. Hence (1) is established. Set (G_{1,2}S)^{+} = G_{1,2}S/Z_{0}, G = G(z), and Ḡ = G/⟨z⟩.

Part (2) follows from the fact that α̃ is the L_{3}(q)-amalgam.

Assume q = 4. Then O^{2}(M̄) = L_{3}(4) or P GL_{3}(4), and in the former case Y = 1, so we may assume the latter, where |Y : T_{0}| = 3 and T_{0}^{+} = [T_{0}^{+}, Y], so that Y is irreducible on Q^{+}. Suppose O_{2}(Y S) ≠ T_{0}. If T ≠ T_{0} then from 11.2.10, the field involution in F inverts Y/T_{0}. Thus case (1) of 10.1 holds, and there is s ∈ O_{2}(Y S) and s^{2} ∈ T_{0}, so by 11.3 we may choose s to be a graph or graph-field involution. Let B ≤ Σ ∈ Syl_{3}(G_{1,2}). As in the proof of 11.2.9, s acts on T_{0}Σ. As O^{2}(M̄) ≅ P GL_{3}(4), Σ = B_{1} × B_{2} where B_{1} = B = Σ ∩ G̃_{1,2} and B_{2} = Σ ∩ Y. Then s acts on B_{i}T_{0} for i = 1, 2, and as s ∈ R, [s, B_{2}] ≤ R. However s is nontrivial on ΣT_{0}/T_{0}, so s inverts B_{1}T_{0}/T_{0}, and hence C_{Z_{0}}(s) = ⟨z⟩, so that s is a graph-field involution. This completes the proof of (3).

Suppose T_{0} ≤ R. As T_{0} = J(S), G ≤ G(T_{0}). From 10.2, G(T_{0}) = G_{1,2}S, so G = C_{G_{1,2}}(z)S = Y S. As α is an M̄-amalgam, G_{1,2} is solvable, so G(z) is solvable. As Z_{0} = Z(T_{0}) ⊴ G(z) and G_{1,2} controls fusion in Z_{0} since T_{0} is weakly closed, Z_{0} is an FT I-subgroup of F by 3.4. Hence (4) holds.

Suppose q > 4. Then T_{0} = O_{2}(Ỹ S) by (2). Further as S ∈ Syl_{2}(G) and S ≤ Ỹ S ≤ G, it follows that R ≤ O_{2}(Ỹ S). Thus R ≤ T_{0}. Then Z_{0} ≤ C_{S}(T_{0}) ≤ C_{S}(R) = Z(R). Suppose Z_{0} ≠ Ω_{1}(R). Then Z_{0} < Q_{i} ∩ R for i = 1 or 2, and then as Ỹ is irreducible on Q_{i}^{+} by (2), it follows that Q_{i} ≤ R. Then in case (1) of 10.1, T_{0} = ⟨Q_{i}^{S}⟩ ≤ R, so indeed R = T_{0}. On the other hand in case (2) of 10.1, Q_{i} is weakly closed in S, so G = C_{G_{i}}(z) ≤ N_{G_{i}}(T_{0}), and once again R = T_{0}.

So assume Z_{0} = Ω_{1}(R). Then Z_{0} ⊴ G, so H = C_{G}(Z_{0}) ⊴ G with R_{H} = C_{R}(Z_{0}) = F^{∗}(H). Then R_{H}^{+} = F^{∗}(H/Z_{0}), so C_{C_{S}(Z_{0})^{+}}(R_{H}^{+}) ≤ R_{H}^{+}. However R_{H}^{+} ≤ T_{0}^{+} ≤ C_{S}(Z_{0})^{+}, with T_{0}^{+} abelian, so T_{0}^{+} ≤ R_{H}^{+}, and again T_{0} = R, completing the proof of (5).

It remains to establish (6)-(8), so we may assume q = 4. Then Z_{0} ≤ Z(S̄) ≤ R̄, so Z_{0} ≤ R.

Assume O^{2}(M̄) = P GL_{3}(4), but T_{0} ≰ R. As in the proof of (5), R ≤ O_{2}(Y S) ≠ T_{0}, so it follows from (3) that case (1) of 10.1 holds and R = R_{0}⟨s⟩, where s is a graph-field involution and R_{0} = T_{0}∩R is proper in T_{0}. Then T_{s} = [T_{0}, s] ≤ R_{0}. Now |T_{s}| = 16 and Y is irreducible on T_{0}/T_{s}, so it follows that R_{0} = T_{s}, and hence |R| = 32 and Y S/R ≅ A_{4} or S_{4}, where T = T_{0} and S is the extension of T_{0} by a graph-field automorphism, or T ≠ T_{0} and M̄ = P ΓL_{3}(4)), respectively. By 11.3, C_{T_{0}}(s) ≅ Q_{8}, so as Z_{0}⟨s⟩ ≅ D_{8}, we conclude that R ≅ Q_{8}D_{8}. Hence Out(R) ≅ O_{4}^{−}(2) ≅ S_{5}. Now Y S/R ≤ Out_{F}(R) ≤ Out(R), O_{2}(Out_{F}(R)) = 1, and S/R is Sylow in Out_{F}(R). We conclude that Out_{F}(R) ≅ A_{5} or S_{5} for T = T_{0} or T ≠ T_{0}, respectively. When T = T_{0}, it follows from 46.8 in [SG] that G ≅ CĜ(z), where Ĝ = J_{3}. Hence the proof of (6) will be complete once we show that T = T_{0} when F is of characteristic 2-type.

We next prove (7), so we may assume O^{2}(M̄) = L_{3}(4). Let R_{!} = O_{2}(G_{1,2}S). Then [R_{!}, T_{0}B] ≤ T_{0}, so either R_{!} = T_{0} or case (1) of 10.1 holds and R_{!} = T_{0}⟨τ⟩, where τ is a graph involution. Let R_{0} = O_{2}(G(Z_{0})); as usual, R_{0} ≤ R_{!}. If R_{0} ≤ T_{0} then an argument in the proof of (5) shows R_{0} = T_{0}. Then G(Z_{0}) = G(R_{0}), so R_{0} = R_{!}. Suppose R_{0} ≰ T_{0}; we claim that again R_{0} = R_{!}. Suppose otherwise; then T_{0} ≰ R_{0}. But for r ∈ R_{0} − T_{0}, r ∈ τT_{0}, so [T_{0}^{+}, τ] = [T_{0}^{+}, r] ≤ R_{0}^{+}, so T_{1} = [T_{0}, τ] ≤ R_{0}. Then as Y is irreducible on T_{0}/T_{1}, it follows that T_{0} ∩ R_{0} = T_{1}. Moreover R_{0}/T_{1} = C_{R_{!}}(Y)/T_{1} = T_{1}⟨τ⟩/T_{1}, so R_{0} = T_{1}⟨τ⟩. However by 11.3.1, T_{0} is transitive on τT_{1}, so each member of τT_{1} is an involution, and hence τ inverts T_{1} ≅ Z_{4}^{2}. If follows that T_{1} is the subgroup of R_{0} generated by elements of order 4, so T_{1} ⊴ G(Z_{0}). Further T_{0} centralizes the factors in the chain 1 < Z_{0} < T_{1} < R_{0}, so T_{0} ≤ R_{0}, completing the proof of the claim. Thus we have shown that O_{2}(G(Z_{0})) = R_{!}, and in particular T_{0} ⊴ G(Z_{0}), so G(Z_{0}) = G_{1,2}S.

If T_{0} ≤ R then G = C_{G(Z_{0})}(z) = S. Thus we may assume T_{0} ≰ R. If Z_{0} ⊴ G then G ≤ G(Z_{0}), contradicting T_{0} ⊴ G(Z_{0}). Therefore Z_{0} is not normal in G.

Let V = ⟨Z_{0}^{G}⟩. Thus there exists an involution v ∈ V − Z_{0}. Suppose first that V is abelian. Then v is an involution in C_{S}(Z_{0}) and C_{S}(Z_{0}) = T_{0} or T_{0}⟨τ⟩, so v is in

Q_{1}^{#} ∪ Q_{2}^{#} ∪ τ^{T_{0}}.

Suppose v ∈ Q_{i}^{#}. Then as V is abelian, V ≤ C_{S}(Z_{0}) ∩ C_{S}(v) = Q_{i}. However in case (1) of 10.1, v^{x} ∈ V −Q_{1}, a contradiction. Thus case (2) of 10.1 holds and [Q_{i}, R] ≤ C_{R}(V) = Q_{i} ∩ R = V or Q_{i}, so Q_{i} ≤ R, and then as usual T_{0} ≤ R, contrary to assumption. Thus we may assume v ∈ τ^{T_{0}}, so that ⟨τ^{T_{0}}⟩ ≤ V, contradicting ⟨τ^{T_{0}}⟩ nonabelian.

Thus we can assume v ∈ S − C_{S}(Z_{0}), so v is a field or graph-field involution. Set V_{v} = ⟨v^{T_{0}}⟩, so that V_{v} ≤ V. If v is a graph-field involution, then we have seen that V_{v} = ⟨v⟩[T_{0}, v] ≅ Q_{8}D_{8}. If v is a field automorphism then [T_{0}, v] = C_{T_{0}}(v)Z_{0} ≅ D_{8} × Z_{2}, so as Z_{0}⟨v⟩ ≅ D_{8}, V_{v} = [T_{0}, v]⟨v⟩ ≅ D_{8}^{2}.

Next Z̄_{0} ≤ Z(S̄), so V̄ ≤ Z(R̄). Therefore R ≤ C_{S}(V̄) = V, so R = V. Now Out(R) ≅ O_{4}^{ϵ}(2), where ϵ = + if v is a field involution, and ϵ = − if v is a graph-field involution. Further Out_{T_{0}}(R) ≅ E_{4}, and we have seen that N_{G}(T_{0}) = C_{G_{1,2}}(z)S = S, so S/R = N_{G/R}(T_{0}R/R). However we also saw that if ϵ = − then Out_{T_{0}}(R) ∈ Syl_{2}(O^{2}(Out(R))), so as O_{2}(G/R) = 1 it follows that O^{2}(Out(R)) ≤ G/R, contradicting S/R = N_{G/R}(T_{0}R/R).

Therefore v is a field involution and G/R ≤ O_{4}^{+}(2). Moreover in case (1) of 10.1, D_{8} ≅ S/R, while in case (2) E_{4} ≅ T_{0}R/R = S/R. As O_{2}(G/R) = 1 and m_{2}(S/R) = 2, we conclude that G/R = Out(R) ≅ O_{4}^{+}(2), or is of index 2 in O_{4}^{+}(2) in cases (1) and (2) of 10.1, respectively. In particular G is transitive on involutions in R − ⟨z⟩. Let T_{v} = C_{T} (v), T_{0,v} = C_{T_{0}}(v), S_{v} = C_{S}(v), P_{i} = C_{Q_{i}}(v), and A_{i} = Aut_{C_{G_{i}} (v)}(P_{i}). Then by 11.2.5, T_{v} = ⟨v⟩ × T_{0,v} with T_{0,v} = P_{1}P_{2} ≅ D_{8}, and A_{i} ≅ S_{3}. Further by 11.3, T_{0} is transitive on the involutions in vT_{0}, so by a Frattini argument, |S_{v} : T_{v}| = |S : T |, so |S_{v}| = 32, 16 in case (1), (2) of 10.1, respectively.

Next as G is transitive on the involutions in R − ⟨z⟩, v^{G} ∩ Z_{0} ≠ ∅. Then as G_{1,2} is transitive on Z_{0}^{#}, v ∈ z^{F}. Thus there is α ∈ 𝔄(v) with vα = z. Now S_{3} ≅ A_{i} ≅ A_{i}α^{∗} ≤ Aut_{G}(P_{i}α), so as R = O^{2′}(O^{2}(G)), if follows that P_{i}α ≤ R. Thus T_{0,v}α = (P_{1}P_{2})α ≤ R, so ⟨zα⟩ = Φ(T_{0,v})α ≤ Φ(R) = ⟨z⟩, contradicting vα = z and v ≠ z. This contradiction completes the proof of (7).

Finally we prove (8). Assume first that T ≠ T_{0}; then we may assume v is a field involution and continue to take S_{v} = C_{S}(v), T_{v} = C_{T} (v), etc. Let α ∈ 𝔄(v). We saw that S_{3} ≅ A_{i}α ≤ Aut_{G}(P_{i}α), so G(vα) is not a 2-group.

Suppose v ∈ z^{F}. Then we may take vα = z. By the previous paragraph, G is not a 2group, so by (7) and our treatment of (6), M̄ = Aut(L_{3}(4)), R ≅ D_{8}Q_{8}, and G/R ≅ S_{5}. As S_{3} ≅ A_{i}α is irreducible on P_{i}α, it follows that P_{i}α ∩R = 1 or P_{i}α ≤ R, and the latter is impossible as m_{2}(R) = 2 while m_{2}(⟨v, P_{i}⟩α) = 3. Therefore for i ∈ I, P_{i}α ∩ R = 1, so as T_{0,v} = P_{1}P_{2}, also T_{0,v}α ∩ R = 1. As G/R ≅ S_{5} and Aut_{G}(P_{i}α) ≅ S_{3}, this is a contradiction.

Therefore v ∉ z^{F}. On the other hand we’ve seen that |S_{v}| = 32, 16, in case (1), (2) of 10.1, and from 11.3, for each involution s ∈ S − T, |C_{S}(s)| ≤ 32. Thus as all involutions in T_{0} are in z^{F}, it follows that v ∈ F^{f}, so we may take vα = v. Let R_{v} = O_{2}(G_{v}) and suppose F is of characteristic 2-type. Then ⟨z, v⟩ = Z(S_{v}) ≤ R_{v}, and hence P_{i} = ⟨z^{A_{i}}⟩ ≤ R_{v}. This is a contradiction as Aut_{P_{1}}(P_{2}) ≰ O_{2}(A_{2}) and A_{2} ≤ Aut_{G}(P_{2}).

Thus we may assume case (2) of Hypothesis 10.1 holds. Let P ∈ F^{frc} and Z_{P} = Ω_{1}(Z(P)) ∩ T_{0}. We’ve shown that z^{F} ⊆ T_{0}, so Z_{P} ⊴ G(P). Further 1 ≠ Z_{0} ∩ Z(P), and we may assume Z_{P} ≤ Q_{1}. Suppose Z_{P} ≤ Z_{0}. By (4), (6), and (7), Z_{0} is an FT Isubgroup of S and G(Z_{0}) ≤ G_{1}, so G(P) ≤ G(Z_{0}) and then G(P) ≤ G_{1}. Thus we may assume Z_{P} ≰ G_{0}, so C_{T_{0}}(Z_{P}) = Q_{1}. Now

[N_{Q_{1}}(P), P] ≤ Q_{1} ∩ P = Ω_{1}(C_{P} (Z_{P})) ∩ T_{0} ⊴ G(P),

so N_{Q_{1}}(P) ≤ P and hence Q_{1} ≤ P. Then as Q_{1} is weakly closed in S, G(P) ≤ G_{1}. This completes the proof of (8) and the lemma.

(11.5) Assume α is the M̄-extension of Sp_{4}(q). Let z be an involution in Z(S), Z_{0} = Z(T_{0}), and for i ∈ I set Z_{i} = Z(K_{i}) and let z_{i} be an involution in C_{Z_{i}}(T). Set Δ = Z_{0} − (Z_{1} ∪ Z_{2}), and in case (2) of 10.1 choose z ∈ Δ. Then for i ∈ I:

(1) N_{F}(Z_{i}) = F_{i}, so G(z_{i}) ≤ G_{i} = G(Z_{i}) and T ∈ Syl_{2}(G(z_{i})).

(2) Z_{i} is a FT I-subgroup of S.

(3) z^{F} ∩ Z_{i} = ∅. Moreover in case (2) of 10.1, Z_{2} ∉ Z_{1}^{F}.

(4) z ∈ Δ and B is regular on Δ.

(5) Each involution in T_{0} is fused into Z_{1}, Z_{2}, or Δ.

(6) T_{0}, Q_{3−i} is the weak closure of Z_{i} in S in case (1), (2) of 10.1, respectively.

(7) G(z) ≤ G_{1,2}S with T ≤ O_{2}(G(z)) and G(z) = SF_{1,2}, where F_{1,2} is cyclic of odd order and induces field automorphisms on K_{1} and K_{2}.

Proof. As Z_{1} and ⟨z_{1}⟩ are normal in X, they are normal in S in case (2) of 10.1, while in case (1) of 10.1, it follows from 7.2.2 that z_{1} and Z_{1} are in F^{f} and T is Sylow in G(Z_{1}) and G(z_{1}). By an argument in the proof of 9.8.4, Q is normal in G(Z_{1}) and G(z_{1}), so by 2.2.5, each is contained in G_{1}. Then as Z_{1} ⊴ G_{1}, (1) holds and Z_{1} ⊴ G(z_{1}). As T_{0} is weakly closed in S, G_{1,2}S = G(T_{0}) controls fusion in Z_{0} = Z(T_{0}). If g ∈ G(T_{0}) and P, P^{g} ≤ Z_{1}, then as Z_{1} ∩ Z_{1}^{h} = 1 for h ∈ G(T_{0}) − G_{1,2}, we have g ∈ G_{1,2} ≤ G_{1}. Thus G_{1} controls fusion in Z_{1}, so Z_{1} is a FT I-subgroup of S by 3.4, establishing (2).

Suppose case (1) of 10.1 holds. Then by (1) and (2), no involution in Z_{1} is fused into Z(S), so (3) holds. For s ∈ S−T, [s, T] ≠ 1, so Z(S) ≤ T. Then Z(S) ≤ Z(T) = C_{Z_{0}}(T). Hence z ∈ Δ by (3).

Suppose on the other hand that case (2) of 10.1 holds. We just saw that G(T_{0}) controls fusion in Z_{0} and acts on Z_{i} for i ∈ I. Moreover in this case, z ∈ Δ by hypothesis. Thus (3) also holds in this case.

From 11.2.5, if b ∈ B^{#} with C_{Z_{0}}(b) ≠ 1, then b ∈ B_{i} for some i ∈ I, and C_{Z_{0}}(b) = Z_{i}. Thus B is semiregular on Δ, so as |Δ| = (q − 1)^{2} = |B|, (4) follows.

As Q_{1} and Q_{2} are the maximal elementary abelian subgroups of T_{0} and each involution in Q_{1} is fused into Z_{1}, Z_{2}, or Δ under G_{1}, (5) follows.

We next prove (6). Let ϕ ∈ hom_{F}(Z_{1}, S) and set P = Z_{1}ϕ. We must show P ≤ T_{0}, so assume otherwise. If P ∩ T_{0} ≠ 1, then by (3) and (5), P ∩ Z_{1}α ≠ 1 for some α ∈ hom_{F}(Z_{1}, T_{0}), so P = Z_{1}α ≤ T_{0} by (2) and 3.8.5, contrary to assumption. Therefore P ∩ T_{0} = 1. But by 11.2.9, T/T_{0} is cyclic, so as |S : T | ≤ 2, we conclude that case (1) of 10.1 holds, e = m_{2}(P) = 2, P ≰ T, and P ∩T ≠ 1. Let 1 ≠ y ∈ P ∩T. Then 1 ≠ C_{Z_{1}}(y). However by (2) and 3.8.8, P is a TI-subgroup of S, so 1 ≠ C_{Z_{1}}(y) ≤ C_{T_{0}}(y) = C_{T_{0}}(P), whereas P ≰ T so C_{Z_{1}}(P) = 1. This contradiction completes the proof of (6).

It remains to prove (7). Let G = G(z) and R = O_{2}(G). Suppose Z_{0} ≤ R. If Z_{0} is weakly closed in R with respect to G, then Z_{0} ⊴ G, so |G : N_{G}(Z_{1})| = |S : T | by (2)-(4). But by (1), N_{G}(Z_{1}) = C_{G_{1}}(z), and by 11.2.5, T ⊴ C_{G_{1}}(z), so T ≤ R. Hence Z_{0} is not weakly closed in R, so by (6), Q_{i} ≤ R for some i ∈ I. Suppose case (1) of 10.1 holds. Then T_{0} = ⟨Q_{i}, Q_{i}^{x}⟩ ≤ R, so as T_{0} is weakly closed in S, G ≤ G(T_{0}) = G_{1,2}S. Then by 11.2.5, T ⊴ G = C_{G_{1,2}}(z)S = F_{1,2}S. So suppose instead that case (2) of 10.1 holds. Then Q_{i} is weakly closed in S, so G ≤ G_{i} and again T ⊴ G = C_{G_{i}}(z) by 11.2.5.

Therefore we may assume Z_{0} ≰ R, and then assume that Z_{1} ≰ R. If R ≤ T_{0} then Z_{0} ≤ C_{S}(R) ≤ R, contrary to assumption. Therefore R ≰ T_{0}.

Set T_{0}^{+} = T_{0}/Q_{1} and R_{0} = R∩T_{0}. Suppose R_{0}^{+} is noncyclic. Then there exists t_{j} ∈ R_{0} for j = 1, 2, with t_{1}^{+} ≠ t_{2}^{+} nontrivial. Now W_{j} = [Q_{1}, t_{j}] is a 1-dimensional F_{q}-subspace of Z_{0} with W_{1} ∩ W_{2} = 0, so as t_{j} ∈ R, also Z_{0} = W_{1}W_{2} ≤ R, a contradiction. Therefore R_{0}^{+} is cyclic.

Suppose s ∈ R − T_{0}. From Q_{1}^{s} = Q_{2}, so T_{0} = [T_{0}, s]Q_{1}. Then as [T_{0}, s] ≤ R_{0}, R_{0}^{+} is noncyclic, a contradiction.

Thus R ≤ T, so as R ≰ T_{0}, T ∩ R ≰ T_{0}. Then for f a field involution, there exists r ∈ fT_{0} ∩ R, and [T_{0}, r]^{+} = [T_{0}, f]^{+}. As [T_{0}, r] ≤ R_{0}, [T_{0}, r]^{+} is cyclic, so q = 4. Now Z_{r} = [T_{0}, r] ∩ Z_{0} is a hyperplane of Z_{0} contained in R, so as Z_{0} ≰ R, Z_{r} = R ∩ Z_{0}. In particular Z_{i} ∩ Z_{r} = ⟨z_{i}⟩ is of order 2 for i = 1, 2, and as ⟨r^{T_{0}}⟩ ≤ R ≤ T, is follows that Z_{r} = Z(R), and then that z_{i}^{F} ∩ Z(R) ⊆ {z_{1}, z_{2}} by (3) and (4). But now G acts on {z_{1}, z_{2}}, and hence T_{0} ≤ R by (1) and (2), for our final contradiction.

(11.6) (1) If F is of characteristic 2-type then T = T_{0}.

(2) If case (2) of Hypothesis 10.1 holds then for each P ∈ F^{frc}, G(P) ≤ G_{1} or G_{2}.

Proof. Assume otherwise. From 11.2.5, α is an M̄-extension of L_{3}(q) or Sp_{4}(q). If q = 4 then by 11.4.8, α̃ is the Sp_{4}(4)-amalgam.

Assume first that T ≠ T_{0}. Then 11.2.9 says there is a field involution f ∈ T acting faithfully as an involutory field automorphism on K_{1} and K_{2}. Hence q = q_{0}^{2} is a square, and setting J_{i} = C_{K_{i}}(f), either J_{i} is an L_{2}(q_{0})-block or q = 4 and J_{i} ≅ Z_{2} × S_{4}.

Let S_{f} = C_{S}(f), T_{f} = C_{T} (f), U_{i} = [O_{2}(J_{i}), J_{i}], A_{i} = Aut_{J_{i}}(U_{i}), and E = [U_{1}, U_{2}]. Then A_{i} ≅ L_{2}(q_{0}), and in particular A_{i} is nonsolvable if q > 4. Let δ ∈ 𝔄(f). Then B_{i} = A_{i}δ^{∗} ≤ Aut_{G(fδ)}(U_{i}δ), so either G(fδ) is not solvable, or q = 4 and G(fδ) is not a 2-group. We conclude from parts (4) and (5) of 11.4, that if α̃ is of type L_{3}(q), then fδ ∉ Z(S), and hence that f^{F} ∩ T_{0} = ∅. Similarly if α̃ is of type Sp_{4}(q), fδ ∉ Δ by 11.5.7.

Suppose α̃ is of type Sp_{4}(q), adopt the notation of 11.5, and assume fδ = z_{1}. By parts (1) and (2) of 11.5, G(z_{1}) ≤ G_{1} and Z_{1} is an FT I-subgroup of S. Hence if q > 4 then B_{i} = B_{i}^{∞} is induced in G(z_{1})^{∞} = K, and as B_{i} is nonsolvable, U_{i}δ ≤ Q_{1}. Similarly if q = 4 then O^{2}(B_{i}) is induced in O^{2}(G(z_{1})) = K, and either U_{i}δ ≤ Q_{1} or U_{i}δ ∩ Q_{1} = 1.

As U_{1}U_{2} is nonabelian and Q_{1} is abelian, there exists i ∈ I with U_{i}δ ≰ Q_{1}. Therefore by the previous paragraph, q = 4 and U_{i}δ ∩ Q_{1} = 1. Therefore Eδ ≰ Q_{1}, so as U_{3−i} = ⟨E^{A_{3−i}}⟩, also U_{3−i}δ ∩ Q_{1} = 1. Set G_{1}^{∗} = G_{1}/Q_{1}. Then G_{1}^{∗} ≅ ΓL_{2}(4) and D_{8} ≅ U_{1}U_{2} ≅ ((U_{1}U_{2})δ)^{∗} with Aut_{G_{1}}(U_{i}δ) ≅ S_{3}, a contradiction.

Therefore if α̃ is of type Sp_{4}(q), then for i ∈ I, f ∉ z_{i}^{F}, and we showed earlier that f^{F} ∩ Δ = ∅, so by parts (4) and (5) of 11.5, f^{F} ∩ T_{0} = ∅. Recall also that we showed earlier that f^{F} ∩ T_{0} = ∅ when α̃ is of type L_{3}(q). Next by 11.3, if s ∈ S − T is an involution then |C_{S}(s)| ≤ |S_{f}|, so f ∈ F^{f}. Thus we may take fδ = f.

Assume that F is of characteristic 2-type and set G = G(f) and R = O_{2}(G). Then 1 ≠ R_{i} = C_{U_{i}}(S_{f}) ≤ R, and then U_{i} = ⟨R_{i}^{A_{i}}⟩ ≤ R. However Aut_{U_{1}}(U_{2}) ≰ O_{2}(A_{2}), a contradiction.

This completes the proof of (1), so we may assume case (2) of Hypothesis 10.1 holds. If α̃ is of type L_{3}(q), then we can repeat the proof of 11.4.8 almost verbatim, so we may assume α̃ is of type Sp_{4}(q).

Let P ∈ F^{frc} and Z_{P} = Ω_{1}(Z(P)) ∩ T_{0}. We showed t^{F} ⊆ T_{0} for each involution t in T_{0}, so Z_{P} ⊴ G(P). Suppose 1 ≠ Z_{i} ∩ Z_{P} for i = 1 or 2. If F_{i} = ⟨z_{i}^{F} ∩ Z_{P} ⟩ ≤ Z_{0} then as Z_{i} is an FT I-subgroup of S by 11.5.2, G(P) ≤ G(Z_{i}) = G_{i}. Thus we may assume F_{i} ≰ Z_{0}, so Z_{P} ≤ Q_{2} and Q_{2} = C_{T_{0}}(Z_{P}). Then arguing as in the proof of 11.4.8, G(P) ≤ G_{2}.

Therefore we may assume Z_{i} ∩ Z_{P} = 1 for i = 1, 2, so Z_{0} ∩ Z_{P}^{#} ⊆ Δ. Then by 11.5.7, either Γ = z^{F} ∩ Z_{P} ⊆ Δ and G(P) ≤ G_{i} for i = 1, 2, or Γ is not contained in Δ but Γ ⊆ Q_{j} for j = 1 or 2, and C_{T_{0}}(Z_{P}) = Q_{j}. If course we may assume the latter. Then, arguing as in the proof of 11.4.8, Q_{j} ≤ P so G(P) ≤ G_{j}, completing the proof of (2) and the lemma.

(11.7) Assume case (2) of 10.1 holds. Then

(1) If F is of characteristic 2-type then S = T_{0} and α is an M̄-amalgam for some extension M̄ of G_{0} = L_{3}(q) or Sp_{4}(q) of odd degree with S ∈ Syl_{2}(M̄).

(2) F = ⟨F_{S}(G_{1}), F_{S}(G_{2})⟩.

(3) F = F_{S}(M̄).

Proof. The first statement in (1) follows from 11.6, and then that statement and 11.2.5 imply the second. Part (2) follow from 11.6.2.

By 11.2.5, the amalgam α is determined up to isomorphism, and by (2) the amalgam α determines F up to isomorphism. Then as F_{S}(M̄) satisfies our hypothesis, (3) follows.

Because of 11.7, in the remainder of this section we assume:

Hypothesis 11.8. Hypothesis 11.1 holds, and we are in case (1) of Hypothesis 10.1.

(11.9) Let M̂ be P ΓL_{3}(q) or ΓSp_{4}(q) for α̃ of type G_{0}, where G_{0} = L_{3}(q), Sp_{4}(q) respectively. Then

(1) Aut(α̃) = AutM̂(α̃) ≅ M̂_{1,2}.

(2) Aut(α) = AutM̂(α) ≅ NM̂_{1,2}(G_{1,2}).

Proof. Let β = α̃ or α, and A = Aut(β). Then β = (β_{i} : H_{1,2} → H_{i} : i ∈ I) for suitable groups H_{J}, J ⊆ I. Set A = Aut(β). Then we have representations ρ_{J} : A → A_{J} = Aut(H_{J}) for ∅ ≠ J ⊆ I. Set W_{J} = ker(ρ_{J}); then W_{1} ∩ W_{2} ∩ W_{1,2} = 1 and W_{i} ≤ W_{1,2} for i ∈ I, so W_{1} ∩ W_{2} = 1. We will see that W_{1,2} ≤ W_{i}, so W_{1} = W_{1,2} = W_{1} = 1; that is ρ_{J} is faithful for each J ⊆ I.

First take β = α̃. Then H_{J} = G̃_{J} is a parabolic in G_{0}, and H_{J} = Q_{J}Y_{J} where Y_{J} is a Levi factor in the parabolic with Y_{1,2} = Y_{1} ∩ Y_{2} and Q_{1,2} = T_{0}. Further H_{J} ≤ M̂_{J} = NM̂(H_{J}) a parabolic in M̂, and M̂_{J} = Q_{J} Ŷ_{J}, where Ŷ_{J} = NM̂_{J} (Y_{J}) is a complement to Q_{J} in M̂_{J}.

Let P_{i} = C_{A_{i}}(Q_{i}). Suppose G_{0} ≠ L_{3}(4). Then O(Y_{i}) ≠ 1, so P_{i} = Q_{i}. Then as N_{GL(Q_{i})}(Y_{i}) = Ŷ_{i}, it follows that Aρ_{i} = M̂_{1,2}. As CM̂_{i}(H_{1,2}) = 1, we have also verified that W_{1,2} ≤ W_{i}, as asserted above. Therefore ρ_{i} : A → M̂_{1,2} is an isomorphism, so (1) holds in this case. Further α̃ is characteristic in α, so (2) follows from (1).

So assume that G_{0} = L_{3}(4). Then A_{i} = P_{i}Ŷ_{i} where P_{i} ≅ E_{64} is an indecomposable Y_{i}-module, and again C_{A_{i}}(G̃_{1,2}) = 1, so indeed W_{1,2} ≤ W_{i}. Thus ϕ = ρ_{1}^{−1}ρ_{2} : A_{1}ρ_{1} → A_{2} is an injection. Suppose u ∈ P_{1} ∩Aρ_{1} −Q_{1} and let v = uϕ. We may choose u to centralize the Cartan group B of H_{1,2}. Then v centralizes Q_{1}B so [H_{2}, v] ≤ Q_{2}. Then H_{2} acts on [Q_{2}, v] = Z(T_{0}), impossible as H_{2} is irreducible on Q_{2}. Thus again P_{i} ∩ Aρ_{i} = Q_{i}, so (1) also holds in this case.

We have verified (1), so we may now take β = α. As α̃ is characteristic in α, A_{J} acts on G̃_{J}. Further [C_{A_{i}}(G̃_{i}), G_{i}] ≤ C_{G_{i}}(G̃_{i}) = 1, so A_{i} is faithful on G̃_{i}. Then (2) follows.

Definition 11.10. Let β = (β_{i} : H_{1,2} → H_{i}) be a extension of α̃. Define a similarity of β to be a family σ = (σ_{J} : ∅ ≠ J ⊆ I) of maps σ_{J} such that either σ is an automorphism of β, or σ_{1,2} ∈ Aut(H_{1,2}) and for i ∈ I, σ_{i} : H_{i} → H_{3−i} is an isomorphism. Of course in addition we demand that for each i ∈ I, β_{i}σ_{i} = σ_{1,2}β_{3−i}.

Compose similarities in the obvious way, and write Aut^{∘}(β) for the group of similarities of β. Thus |Aut^{∘}(β) : Aut(β)| = 2. For α̃ of type G_{0} = L_{3}(q) or Sp_{4}(q), let M_{0} = Aut(G_{0}), and for J ⊆ I, set M_{J} = N_{M_{0}}(G̃_{J}). Write M̂ = N_{M_{0}}(G̃_{1}^{G_{0}}) as in 11.9; thus |M_{0} : M̂| = 2.

(11.11) Adopt the notation of 11.10 and define π : M_{1,2} → Aut^{∘}(α̃) by decreeing that for m ∈ M_{1,2}, mπ_{J} is conjugation by m. Then

(1) π : M_{1,2} → Aut^{∘}(α̃) is an isomorphism.

(2) Aut^{∘}(α) = Aut_{M_{1,2}}^{∘}(α) ≅ N_{M_{1,2}}(G_{1,2}).

(3) Let c = (c_{J} : J ⊆ I) be the similarity of α defined in 10.2. Then Σ = ⟨G_{1,2}π, c⟩ ≤ Aut^{∘}(α) with |Σ : G_{1,2}π| = 2 and c^{2} = x^{2}π. Further writing ρ : Σ → Aut(G_{1,2}) for the restriction representation of Σ on G_{1,2}, and γ : G(T_{0}) → Aut(G_{1,2}) for the conjugation map, ρ and γ are faithful with Σρ = G(T_{0})γ, πρ = γ on G_{1,2}, and cρ = xγ.

(4) There is an injective group homomorphism ξ : G(T_{0}) → N_{M_{1,2}}(G_{1,2}) which is the identity on G_{1,2}.

(5) If F is of characteristic 2-type and α̃ is of type Sp_{4}(q) then e is odd.

Proof. Let β = α̃ or α; thus β = (β_{i} : H_{1,2} → H_{i} : i ∈ I). By 11.2.5, we may take β_{i} to be inclusion.

First take β = α̃. For m ∈ M_{1,2} − M̂ and h ∈ H_{1,2},

hβ_{i}(mπ_{i}) = h(mπ_{i}) = h^{m} = (h^{m})β_{3−i} = h(mπ_{1,2})β_{3−i},

and similarly for m ∈ M_{1,2} ∩ M̂, β_{i}(mπ_{i}) = (mπ_{1,2})β_{i}, so π : M_{1,2} → Aut^{∘}(β). Visibly π is a homomorphism, and as C_{M_{1,2}}(H_{1,2}) = 1, π is an injection.

By 11.9.1, π : M̂ → Aut(β) is an isomorphism, so as |Aut^{∘}(β) : Aut(β)| = 2 = |M_{1,2} : M̂_{1,2}|, (1) follows.

Similarly (1) and 11.9.2 imply (2).

Next we prove (3). By definition of π, πρ = γ on G_{1,2}, and by (2), G_{1,2}π ≤ Aut(α). By construction of c in 10.2, c ∈ Aut^{∘}(α) − Aut(α) and c acts on G_{1,2}π with cρ = xγ, and c^{2} = x^{2}π. Thus Σ ≤ Aut^{∘}(α) with |Σ : G_{1,2}π| = 2. Thus (3) holds.

By (1) and (2), π : N_{M_{1,2}}(G_{1,2}) → Aut^{∘}(α) is an isomorphism, and by (3), ζ = γρ^{−1} : G(T_{0}) → Σ is an isomorphism of G(T_{0}) with a subgroup Σ of Aut^{∘}(α), such that ζ = π on G_{1,2}. Therefore ξ = ζπ^{−1} is an injective group homomorphism of G(T_{0}) into N_{M_{1,2}}(G_{1,2}), which is the identity map on G_{1,2}. This proves (4).

Finally assume F is of characteristic 2-type and G_{0} is Sp_{4}(q). Then M_{0}/G_{0} is cyclic with N_{G_{0}}(G_{1,2}) = G̃_{1,2}, so if follows from (4) that G(T_{0})/G̃_{1,2} is cyclic. In particular a Sylow 2-subgroup S/T_{0} of G(T_{0})/G̃_{1,2} is cyclic, so if e is even then x^{2} induces a nontrivial field automorphism on G_{0}, and hence x^{2} ∉ T_{0}, contrary to x^{2} ∈ T and T = T_{0} from 11.6.1. This establishes (5).

(11.12) Assume α is an M̄-extension of G_{0} = L_{3}(q) or Sp_{4}(q). Set Z = Ω_{1}(Z(S)). Then

(1) 𝔐_{+}(F) = ∅.

(2) 𝔐_{Ω,−}(F) = {X, X^{x}}.

(3) Either

(i) M̄ = P GL_{3}(4) and G(Z) is not solvable. Moreover if F is of characteristic 2-type then G(Z) is isomorphic to the centralizer of an involution in J_{3}.

(ii) G(Z) is solvable and G(Z) ≤ G(T_{0}).

(4) If G(Z) is not solvable then F = ⟨F_{S}(G(Z), F_{T} (G_{1})⟩.

(5) If G(Z) is solvable then F = ⟨F_{S}(G(T_{0})), F_{T} (G_{1})⟩.

Proof. Assume X′ ∈ 𝔐_{+}(F) and let V = Ω_{1}(Z(O_{2}(X′))). By 8.3, V ≤ A ∈ A(S) and there exists A′ ∈ A(S) with [V, A′] ≠ 1. But by parts (3) and (5) of 9.10, A(S) = {Q_{1}, Q_{2}}. Therefore we may assume V ≤ Q_{1}. Then as S ≤ X′ and X′ acts on V, V ≤ Q_{2} = Q_{1}^{x}. Hence J(S) = Q_{1}Q_{2} ≤ C_{S}(V), contradicting [V, A′] ≠ 1. This establishes (1).

Next assume X′ ∈ 𝔐_{Ω,−}(F). Then by parts (3) and (5) of 9.10 and 11.2.5, O_{2}(X′) ∈ A(S) and X′ = G(O_{2}(X′))^{∞}T. Therefore as A(S) = {Q_{1}, Q_{2}}, X′ = G_{i}^{∞}T ∈ {X, X^{x}}, establishing (2).

Part (3) follows from 11.4 and 11.5.

By Theorem 7.12,

(*)  F = ⟨C_{F}(Z), N_{F}(T_{0}), F_{S(X′)}(X′) : X′ ∈ 𝔐_{+}(F) ∪ 𝔐_{Ω,−}(F).

Then by (1), (2), and (*), and as X ≤ G_{1},

(**)  F = {F_{S}(G(Z)), F_{S}(G(T_{0})), F_{T} (G_{1})⟩.

Then (5) follows from (3) and (**).

Thus we may assume (3i) holds. Recall that G(T_{0}) = G_{1,2}S with G_{1,2} ≤ G_{1} and S ≤ G(Z). Then (4) follows from (**).

(11.13) Assume F of characteristic 2-type. Then F = F_{S}(G) for a finite group G with S ∈ Syl_{2}(G), satisfying one of the following:

(1) α is the P GL_{3}(4)-amalgam, G(Z(S)) is not solvable, and G ≅ J_{3}.

(2) F^{∗}(G) ≅ L_{3}(q), |O^{2}(G) : F^{∗}(G)| is odd, and G is the extension of O^{2}(G) by a graph or graph-field automorphism.

(3) F^{∗}(G) ≅ Sp_{4}(q) with q = 2^{e} and e odd, |G : O^{2}(G)| = 2, and O^{2}(G) is an extension of F^{∗}(G) by a group of field automorphisms of odd order.

Proof. As F is of characteristic 2-type, T = T_{0} by 11.6.1.

Assume first that 11.12.3.i holds, and let z be the involution in Z(S). Then G(z) = C_{G}(z) for some finite group G isomorphic to J_{3}. In particular S ∈ Syl_{2}(G), and F_{S}(G(z)) = F_{S}(C_{G}(z)). Next by 11.12.4,

(*)  F = ⟨F_{S}(G(z)), F_{T_{0}}(G_{1})⟩.

As F_{S}(G) satisfies our hypothesis, also

(**)  F_{S}(G) = ⟨F_{S}(C_{G}(z)), F_{T_{0}}(Y)⟩,

where Y = N_{G}(Q). Let B_{Z} ∈ Syl_{3}(C_{G_{1}}(Z_{0})) centralize B; thus E = T_{0}B_{Z} = C_{G_{1}}(z). Form the amalgam j = (j_{1} : E → G_{1}, j_{2} : E → G(z)), where j_{i} is the inclusion map. Similarly form the amalgam ι = (ι_{1} : E → Y, ι_{2} : E → C_{G}(z)), where ι_{i} is the inclusion map. From 11.2.5, there exists an isomorphism μ : G_{1} → Y with Eμ = E. We claim that we can choose μ so that μ is the identity on E. If so, then F_{T_{0}}(G_{1}) = F_{T_{0}}(Y), and hence F = F_{S}(G) by (*) and (**), so (1) holds in this case.

So to complete the treatment in this case, it remains to establish the claim. Let η = μ_{|E}; then η ∈ Aut(E). We need γ ∈ Aut(Y) such that ηγ_{|E} = 1. Then we replace μ by μγ to obtain the desired map.

As Q_{1}μ = Q_{1}, η acts on Q_{1}. Then as A(T_{0}) = {Q_{1}, Q_{2}}, η also acts on Q_{2}. Adjusting by a field automorphism of Y, we may assume η centralizes E/T_{0}. Then replacing η by ηc_{t} for suitable t ∈ T_{0}, we may assume η centralizes B_{Z}. Next replacing η by ηc_{b} for suitable b ∈ B, we may assume [Q_{2}, η] ≤ Z_{0}. Then as η acts on the complement P = [Q_{2}, B_{Z}] to Z_{0} in Q_{2}, it follows that [P, η] = 1. Replacing η by ηc_{a} for suitable a ∈ C_{BB_{Z}} (P), we may assume [Z_{0}, η] = 1. Hence η centralizes Q_{2}. Therefore [η, E] ≤ C_{E}(Q_{2}) = Q_{2}, so η is a 2-element. Finally η acts on the complement [Q_{1}, B_{Z}] to Z_{0} in Q_{1}, so η centralizes Q_{1}, and hence also E. This completes the proof of the claim, and our treatment of this case.

Therefore we may assume that 11.12.3.ii holds. Hence by 11.12.5:

(a) F = ⟨F_{S}(G(T_{0})), F_{T_{0}}(G_{1})⟩.

Now α is an M̄-amalgam for some extension M̄ of G_{0} = L_{3}(q) or Sp_{4}(q) contained in M̂. By 11.11.4, we may assume G_{1,2} = M̄_{1,2} and G(T_{0}) ≤ M_{1,2}. Let ι be the amalgam ι = (ι_{1} : G_{1,2} → G_{1}, ι_{2} : G_{1,2} → G(T_{0})), where ι_{1} and ι_{2} are the inclusion maps. From (a), the amalgam ι determines F.

Set G = G_{0}G(T_{0}) ≤ M_{0}. Then F_{S}(G) satisfies the hypothesis of F, and has the same amalgam ι, so F_{S}(G) also satisfies (a) and hence

F = ⟨F_{S}(G(T_{0})), F_{T_{0}}(G_{1})⟩ = F_{S}(G).

Next if G_{0} = Sp_{4}(q), then e is odd by 11.11.5, so |G : O^{2}(G)| is odd, and O^{2}(G) is the extension of G_{0} by a group of field automorphisms of odd order. Hence (3) holds in this case.

Finally assume G_{0} = L_{3}(q). By 11.6, T = T_{0}, so S = T_{0}⟨s⟩ where s is an involution inducing a graph or graph-field automorphism on G_{0}. Further M_{0}/G_{0} is 2-nilpotent, so |O^{2}(G) : G_{0}| is odd. Hence (2) holds in this case, completing the proof.

Section 12. Proofs of the theorems

(12.1) Assume F is a saturated fusion system on a finite 2-group S which is a local CK-system of characteristic 2-type. Assume further that 𝔐_{−}(F) ≠ ∅. Then F is an obstruction to pushing up at the prime 2.

Proof. As F is of characteristic 2-type and 𝔐_{−}(F) ≠ ∅, Hypothesis 9.9 is satisfied with Ω = ∅. If X ≅ S_{4}, then by 9.12, F appears in one of the first three case in the list of obstructions to pushing up in the introduction. If X ≅ S_{4} × Z_{2} then by 9.14, F appears in case (4) of that list. Hence by parts (1) and (5) of 9.10, we may assume K is an L_{2}(q)-block, where q = 2^{e} and e > 1. Thus Hypothesis 10.1 is satisfied, so by 10.3.2, also Hypotheses 11.1 and 11.8 are satisfied. Now by 11.13, F appears in on of cases (5)-(7) of the list of obstructions, completing the proof of the lemma.

We are now in a position to prove Theorem 1. Assume F satisfies the hypothesis of that theorem but does not satisfy conclusion (1) of the theorem. Then by Corollary 7.13 and the definition of B(F) in Definition 7.11, 𝔐_{−}(F) is nonempty. Now 12.1 completes the proof of Theorem 1.

(12.2) Assume G is a finite group of characteristic 2-type with Sylow 2-subgroup S. Then one of the following holds:

(1) m_{2}(G) = 1 and G = O(G)C_{G}(z), where z is the involution in S. Further S = C_{G}(z), or C_{G}(z)) ≅ SL_{2}(3) or SL_{2}(3)^{(2)}.

(2) G is almost simple.

(3) F^{∗}(G) = O_{2}(G).

Proof. This is an easy exercise, given the Z^{∗}-Theorem [Gl1] or the theorem of Brauer and Suzuki on groups with quaternion Sylow 2-subgroups. [BS]

We next prove Theorem 2. Assume G is a counter example to Theorem 2. Suppose first that conclusion (1) of 12.2 is satisfied. Then C_{G}(Z(S)) controls fusion in S, so that conclusion (1) of Theorem 2 holds, contrary to the choice of G as a counter example. Similarly if conclusion (3) of 12.2 is satisfied then N_{G}(O_{2}(G)) controls fusion, for the same contradiction. Therefore by 12.2, G is almost simple.

Set L = F^{∗}(G) and F = F_{S}(G). As G is of characteristic 2-type, F is of characteristic 2-type. As all 2-locals of G are K-groups, F is a local CK-system. Therefore F satisfies Hypothesis 8.1, so by Theorem 1, F is an obstruction to pushing up at the prime 2.

Therefore F = F_{S}(Ḡ) for some finite group Ḡ described in the list of obstructions appearing in the introduction.

Suppose first that S is dihedral of order at least 16. Then we conclude from the classification of almost simple groups of characteristic 2-type with such Sylow 2-subgroups (cf. Lemma I.4.3 of [ASm]) that conclusion (2) of Theorem 2 is satisfied, contrary to the choice of G.

Suppose next that S is semidihedral. Then S ∩ L is dihedral or semidihedral. In the first case as L is of characteristic 2-type, L ≅ L_{2}(p) with p a Fermat or Mersenne prime, or L ≅ A_{6}. (cf. I.4.3 in [ASm]) Then as S is semidihedral, it follows that G ≅ M_{10}, so (3) holds.

So assume S ∩ L is semidihedral. Then we conclude from the classification of simple groups of characteristic 2-type with semidihedral Sylow 2-subgroups (cf. I.4.3 in [ASm]) that conclusion (3) of Theorem 2 is satisfied, for the usual contradiction.

Next assume F appears in case (4) on the list of obstructions. If Ḡ is Aut(A_{6}) then O^{2}(Ḡ) ≅ A_{6} has dihedral Sylow 2-groups of order 8, so S ∩ L ≅ D_{8}. As L is of characteristic 2-type, it follows from the classification of such simple groups (cf. I.4.3 in [ASm]) that L ≅ A_{6} or L_{3}(2). Then G ≅ Aut(A_{6}), a contradiction as Aut(A_{6}) is not of characteristic 2-type.

So suppose Ḡ = Aut(L_{3}(3)). Then S ∩ L is semidihedral of order 16, so as above, L ≅ L_{3}(3) or M_{11}. This forces G = Aut(L), so conclusion (4) of Theorem 2 is satisfied, contrary to the choice of G.

Assume F appears in case (5) of the list of obstructions, so that Ḡ ≅ J_{3}. Let ⟨z⟩ = Z(S). As G is of characteristic 2-type, F_{S}(C_{G}(z)) and F_{S}(CḠ(z)) are models for C_{F}(z)), so it follows that C_{G}(z) ≅ CḠ(z). Further the fusion of involutions in the two groups is the same, so G has one class of involutions. It follows (cf. Lemma I.4.7 in [ASm] and the discussion in its proof) that G ≅ J_{3}, so that conclusion (5) of Theorem 2 is satisfied.

Finally assume F appears in case (6) or (7) of the list of obstructions. Then T_{0} = J(S) is Sylow in O^{2}(Ḡ) and hence also in L_{0} = O^{2}(G). Further as L_{0} is of characteristic 2type, the model N_{L_{0}}(Q_{i}) for N_{F}(Q_{i}) is isomorphic to the model N_{O^{2}(Ḡ)}(Q_{i}). Observe the amalgam α = (α_{i} : G_{1,2} → G_{i} : i ∈ I) is realized in L_{0} with α_{i} the inclusion maps. Namely as L_{0} is of characteristic 2-type, N_{L_{0}}(T_{0}) = G_{1,2} and N_{L_{0}}(Q_{i}), i ∈ I are models for N_{O^{2}(F)}(T_{0}) and N_{O^{2}(F)}(Q_{i}), respectively. Now we can verify that α, L_{0}, and M_{0} = ⟨N_{L_{0}}(Q_{i}) : i ∈ I⟩ satisfy the hypothesis of Theorem F.4.31 in [ASm] in the roles of “M”, “M_{0}”, and γ. Namely by construction, G_{i} = N_{L_{0}}(Q_{i}), so hypothesis (a) of F.4.31 in [ASm] is satisfied. Hypothesis (b) is satisfied as O^{2}(G(z_{1})) ≤ G_{1} for z_{1} ∈ Z(K) when F^{∗}(Ḡ) ≅ Sp_{4}(q), and for z_{1} ∈ Z(S) when F^{∗}(Ḡ) ≅ L_{3}(q), and since O^{2}(G(z_{1})) ≅ O^{2}(C_{L_{0}}(z)). Similarly hypothesis (c) follows by symmetry between G_{1} and G_{2}, and 11.5.7. Hypothesis (d) follows as G_{1,2} contains the Sylow 2-subgroup T_{0} of L_{0}. Hypothesis (e) follows as L_{0} = O^{2}(L_{0}) by construction.

As the hypothesis of F.4.31 in [ASm] is satisfied, that theorem says that L_{0} is isomorphic to an extension of F^{∗}(Ḡ) ≅ L_{3}(q) or Sp_{4}(q) of odd degree. Thus L ≅ L_{3}(q) or Sp_{4}(q). Further S = ⟨s⟩T_{0}, where s is an involution inducing an outer automorphism on L, and s is a graph or graph-field automorphism if L is L_{3}(q). But now F^{∗}(C_{L}(s)) ≠ O_{2}(C_{L}(s)), contradicting G of characteristic 2-type. Thus the proof of Theorem 2 is complete.

We next prove Corollary 3. Let B = ⟨N_{F}(V) : V ∈ U⟩. As U is the greatest member of U, B = N_{F}(U). Thus if F = B then (1) holds, so we may assume B ≠ F. Hence by Theorem 1, F is an obstruction to pushing up at the prime 2.

It remains to inspect the list of obstructions to determine when U has a greatest member U, and to determine U in those cases. For example in cases (6) and (7) on the list, for V ∈ U, 1 ≠ V_{0} = V ∩ T_{0} and as T_{0} is Sylow in L = F^{∗}(G), V_{0} ⊴ N_{G}(V). Then as T_{0} ≤ N_{L}(V_{0}), N_{L}(V_{0}) is one of the three proper parabolics G̃_{1}, G̃_{2}, G̃_{1,2} over T_{0}. As G̃_{1}^{s} = G̃_{2} for s ∈ S − T_{0}, it follows that N_{L}(V) = G̃_{1,2}, and then that V = O_{2}(N_{G}(T_{0})). Hence conclusion (6) or (7) of Corollary 3 holds in this case.

If F appears in case (5) of the list, then F = F_{S}(G) with G ≅ J_{3}. From 11.13 and its proof, U has two maximal members: O_{2}(C_{F}(Z(S)) and T_{0}, a contradiction.

The analysis of the remaining cases is straightforward, and left to the reader. Thus the proof of Corollary 3 is complete.

We next prove Corollary 4, so assume G is a counter example to that result. Appealing to 12.2 and arguing as in the proof of Theorem 2, either conclusion (1), (2), or (3) of Corollary 4 holds, or G is almost simple, and as G is a counter example to Corollary 4, the latter must hold. Thus L = F^{∗}(G) is a nonabelian simple group.

Again arguing as in the proof of Theorem 2, F = F_{S}(G) satisfies Hypothesis 8.1. Set U = O_{2}(M). As M is the unique maximal 2-local subgroup containing S, it follows that for each V ∈ U, V ≲ U, so the hypothesis of Corollary 3 is satisfied.

Suppose first that conclusion (1) of Corollary 3 is satisfied, so that U ⊴ F. Let A = Z(U). As U ⊴ F, M controls fusion in S, so A is a nontrivial strongly closed abelian subgroup of S. Therefore as G is almost simple, Goldschmidt’s Fusion Theorem [Go] says that L is a Goldschmidt group. Then as G is of characteristic 2-type, conclusion (7) of Corollary 4 holds, contrary to the choice of G.

Therefore G satisfies one of conclusions (2)-(7) of Corollary 3, and, from the proof of Corollary 3, F is an obstruction to pushing up. Then from Theorem 2, G satisfies one of conclusion (2)-(5) of Theorem 2. It remains to compare the fusion systems F_{S}(G), for G on the latter list of groups, to the fusion systems on the former list. We conclude that G satisfies one of conclusions (4)-(6) of Corollary 4, completing the proof of Corollary (4).

(12.3) Assume F is a saturated fusion system on a finite 2-group S which is a local CKsystem of Baumann characteristic 2. Assume M ∈ 𝔐_{+}(F). Then G(F) ≠ ∅, so there exists Campbell pairs, and for each Campbell pair (C_{1}, C_{2}) for F, one of the following holds:

(1) C_{1} ≤ Z(M).

(2) C_{2} ⊴ M.

(3) there exists a finite group G with S ∈ Syl_{2}(G) such that F = F_{S}(G) and either G is M_{23} or G is an extension of L ≅ L_{3}(2^{e}) or Sp_{4}(2^{e}) by a group trivial on the Dynkin diagram of L.

Proof. As 𝔐_{+}(F) ≠ ∅, there exists H ∈ G(F) from the definition of G(F) in Definition 8.8, and the discussion in 8.4, and then from Definition 8.8, Campbell pairs exist.

Suppose (1) and (2) fail. Then from Definition 8.5, M ∈ 𝔐(F, C_{1}, C_{2}). Then as F is a local CK-system of Baumann characteristic 2, it follows from 8.13 that Hypothesis 9.7 holds. By 8.14, K_{1} = O^{2}(H_{1}) is an A_{3}-block or L_{2}(q)-block for some q = 2^{e} > 2. In the first case (3) holds by 9.15, so we may assume that K_{1} is an L_{2}(q)-block. Thus case (2) of Hypothesis 10.1 is satisfied. If case (2b) of 10.3 holds then case (3) of 12.3 holds by 10.4. Then by 10.3 we may assume conclusion (2a) of 10.3 holds, so Hypothesis 11.1 holds. Therefore (3) holds by 11.7.4, completing the proof.

(12.4) Assume F is a saturated fusion system on a finite 2-group S which is a local CK-system of characteristic 2-type. Assume 𝔐_{+}(F) ≠ ∅, and 𝔐_{−}(F) = ∅. Then G(F) ≠ ∅, so there exist Campbell pairs, and either

(1) for each Campbell pair (C_{1}, C_{2}) for F, F = ⟨C_{F}(C_{1}), N_{F}(C_{2})⟩, or

(2) there exists a finite group G with S ∈ Syl_{2}(G) such that F = F_{S}(G) and either G is M_{23} or G is an extension of L_{3}(2^{e}) or Sp_{4}(2^{e}) by a group of odd order for some positive integer e.

Proof. As 𝔐_{+}(F) ≠ ∅, Campbell pairs ρ = (C_{1}, C_{2}) exit by 12.3. Suppose (1) fails for the pair ρ. Then as 𝔐_{−}(F) = ∅, it follows from 8.13 that there exists M ∈ 𝔐(F, C_{1}, C_{2}). In particular for this choice of M, neither conclusion (1) or (2) of 12.3 is satisfied, so conclusion (3) holds. Then conclusion (2) of this lemma holds by 11.7.1, since F is of characteristic 2-type.

We next prove Theorem 5. Assume the hypothesis of Theorem 5; we may assume that conclusion (4) of the theorem fails. Then by 12.1, 𝔐_{−}(F) = ∅. Hence if 𝔐_{+}(F) ≠ ∅, then conclusion (2) or (3) of Theorem 5 holds by 12.4. Thus we may assume 𝔐_{+}(F) is also empty. Finally by 8.6, (C_{1}, C_{2}) is a Baumann pair, where C_{1} = Ω_{1}(Z(S)) and C_{2} = Baum(S). Then as 𝔐_{−}(F) = 𝔐_{+}(F) = ∅, conclusion (1) of the theorem holds by 8.7 and the definition of B(F, C_{1}, C_{2}) in Definition 8.5. This completes the proof of Theorem 5.

(12.5) Assume F is a saturated fusion system on a finite 2-group S which is a local CK-system of Baumann characteristic 2. Assume F = ⟨N_{F}(U) : 1 ≠ U ⊴ S⟩. Then one of the following holds:

(1) F = ⟨C_{F}(Ω_{1}(Z(S))), N_{F}(Baum(S)⟩.

(2) G(F) ≠ ∅ and for each Campbell pair (C_{1}, C_{2}) for F, F = ⟨C_{F}(C_{1}), N_{F}(C_{2})⟩.

(3) there exists a finite group G with S ∈ Syl_{2}(G) such that F = F_{S}(G) and either G is M_{23} or G is an extension of L ≅ L_{3}(2^{e}) or Sp_{4}(2^{e}) by a group trivial on the Dynkin diagram of L.

Proof. Let 1 ≠ U ⊴ S and N = N_{F}(U). By hypothesis, F is of Baumann characteristic 2, so N is constrained, and hence we can form G = G(U). By 2.5, N is of characteristic 2 type and 𝔐_{−}(N) = ∅, so by Theorem 7.12, F_{S}(G) = ⟨C_{N} (Z), N_{N} (Baum(S)), F_{S}(M) : M ∈ 𝔐_{+}(N)⟩. Thus if for each such U, 𝔐_{+}(N) = ∅, then (1) holds, so we may assume otherwise. Thus there is M ∈ 𝔐_{+}(F). Now (2) or (3) holds by 12.3.

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